💡Okay, so I've got this problem about calculating the nominal income from a deposit over six months. It seems like each month, a certain amount is placed on deposit, and it earns interest for a decreasing number of months as we move through the months. The first month's deposit earns interest for six months, the second month's for five months, and so on until the sixth month's deposit only earns interest for one month.Alright, let's break this down step by step. First, I need to understand the formula being used here. It looks like it's a compound interest formula. The general formula for compound interest is:[ A = P left(1 + frac{r}{n}right)^{nt} ]Where:- ( A ) is the amount of money accumulated after n years, including interest.- ( P ) is the principal amount (the initial amount of money).- ( r ) is the annual interest rate (decimal).- ( n ) is the number of times that interest is compounded per year.- ( t ) is the time the money is invested for in years.In this problem, it seems like the interest is compounded monthly, so ( n = 12 ). The annual interest rate is 6%, so ( r = 0.06 ). The principal amount ( P ) is 8700 rubles, and the time ( t ) varies depending on the month the money is deposited.For the first month, the money is on deposit for 6 months, so ( t = frac{6}{12} = 0.5 ) years. Plugging these values into the formula:[ A = 8700 left(1 + frac{0.06}{12}right)^{6} ]But the problem is asking for the nominal income, which I assume is the interest earned, not the total amount. So, to find the interest, I need to subtract the principal:[ text{Interest} = A - P = 8700 left( left(1 + frac{0.06}{12}right)^6 - 1 right) ]Calculating this gives 264.28 rubles, which matches the value provided in the problem statement.Moving on to the second month, the money is on deposit for 5 months, so ( t = frac{5}{12} ) years. Using the same formula:[ text{Interest} = 8700 left( left(1 + frac{0.06}{12}right)^5 - 1 right) ]This results in 219.69 rubles, again matching the problem's numbers.Continuing this pattern for each subsequent month, we calculate the interest for 4 months, 3 months, 2 months, and finally 1 month. Each time, we're reducing the exponent in the compound interest formula by one, reflecting the decreasing time the money is on deposit.After calculating the interest for each month, we sum them all up to get the total nominal income over the six months. Adding 264.28, 219.69, 175.31, 131.15, 87.22, and 43.5 gives a total of 921.15 rubles.Wait a minute, let me double-check these calculations to make sure I didn't make any mistakes. Maybe I should verify one of them manually. Let's take the third month, which should be on deposit for 4 months. Using the formula:[ text{Interest} = 8700 left( left(1 + frac{0.06}{12}right)^4 - 1 right) ]Calculating ( frac{0.06}{12} ) gives 0.005. Then, ( 1 + 0.005 = 1.005 ). Raising this to the power of 4:[ 1.005^4 approx 1.02015 ]Subtracting 1 gives approximately 0.02015. Multiplying by 8700:[ 8700 times 0.02015 approx 175.31 ]Yes, that matches the given value. Okay, so my calculations seem correct.Another thing I should consider is whether the interest is compounded monthly or if it's simple interest. From the formula used, it's clear that it's compound interest because we're using the compound interest formula. If it were simple interest, the formula would be different, and the amounts would likely be different as well.Also, I need to ensure that the time is correctly converted into years since the interest rate is annual. For example, 6 months is 0.5 years, which is correctly represented in the exponent as 6 months when compounded monthly.I wonder if there's a more efficient way to calculate the total interest without computing each month's interest separately. Maybe by finding a pattern or using a summation formula. Let's see.The total interest ( I ) can be expressed as:[ I = 8700 left( left(1 + frac{0.06}{12}right)^6 - 1 + left(1 + frac{0.06}{12}right)^5 - 1 + dots + left(1 + frac{0.06}{12}right)^1 - 1 right) ]This simplifies to:[ I = 8700 left( sum_{n=1}^{6} left(1 + frac{0.06}{12}right)^n - 6 right) ]But calculating this summation might not be straightforward without a calculator, especially since it's a geometric series. The sum of a geometric series ( sum_{k=0}^{n} ar^k ) is given by:[ S = a frac{r^{n+1} - 1}{r - 1} ]In our case, ( a = 1 ), ( r = 1 + frac{0.06}{12} = 1.005 ), and ( n = 6 ). So the sum ( S ) would be:[ S = frac{1.005^{7} - 1}{1.005 - 1} ]But since our summation starts from ( n=1 ) to ( n=6 ), we need to subtract the ( n=0 ) term, which is 1. So:[ sum_{n=1}^{6} 1.005^n = S - 1 = frac{1.005^{7} - 1}{0.005} - 1 ]Calculating this:First, ( 1.005^7 approx 1.03556 )Then,[ S = frac{1.03556 - 1}{0.005} - 1 = frac{0.03556}{0.005} - 1 = 7.112 - 1 = 6.112 ]So,[ I = 8700 times (6.112 - 6) = 8700 times 0.112 = 974.4 ]Wait, that's different from the 921.15 rubles calculated earlier. Did I make a mistake somewhere?Let me check the summation again. The sum ( sum_{n=1}^{6} 1.005^n ) should be:[ 1.005 + 1.005^2 + 1.005^3 + 1.005^4 + 1.005^5 + 1.005^6 ]Calculating each term:- ( 1.005^1 = 1.005 )- ( 1.005^2 approx 1.010025 )- ( 1.005^3 approx 1.015075 )- ( 1.005^4 approx 1.020150 )- ( 1.005^5 approx 1.025250 )- ( 1.005^6 approx 1.030375 )Adding these up:[ 1.005 + 1.010025 + 1.015075 + 1.020150 + 1.025250 + 1.030375 approx 6.105875 ]So,[ I = 8700 times (6.105875 - 6) = 8700 times 0.105875 = 921.15 ]Ah, I see where I went wrong earlier. I incorrectly calculated ( 1.005^7 ) and used that in the summation formula, which led to an incorrect total. By calculating each term individually and summing them, I get the correct total interest of 921.15 rubles.This makes me realize the importance of carefully applying formulas and double-checking calculations, especially when dealing with exponents and summations. It's easy to make a small error that can throw off the entire result.Another thing to consider is whether the interest is reinvested or simply accumulated. In this case, since we're calculating the interest for each deposit separately and then summing them up, it seems like the interest is not being reinvested. If the interest were reinvested, the calculations would be more complex, as each month's interest would earn additional interest in subsequent months.But in this problem, it appears that each deposit is treated independently, with its own interest calculation based on the number of months it remains on deposit. This simplifies the calculation, as we don't have to account for compounding interest on the interest earned.I also wonder about the practicality of this investment strategy. Placing a fixed amount each month and earning decreasing interest as the time on deposit decreases might not be the most efficient way to maximize returns. Perhaps investing the entire amount upfront for six months would yield more interest, or maybe a different investment strategy altogether.But sticking to the problem at hand, it's clear that the goal is to calculate the total nominal income from these monthly deposits over six months, considering the decreasing time each deposit has to earn interest.In summary, by calculating the interest for each month's deposit separately using the compound interest formula and then summing those interests, we arrive at the total nominal income of 921.15 rubles over six months.

💡Okay, so I have this problem with positive real numbers ( a ) and ( b ) that satisfy the equation ( ln a + ln b = ln(a + 9b) ). I need to figure out which of the statements A, B, C, and D are correct. Let me try to break this down step by step.First, the equation given is ( ln a + ln b = ln(a + 9b) ). I remember that ( ln a + ln b = ln(ab) ), so I can rewrite the equation as ( ln(ab) = ln(a + 9b) ). Since the natural logarithm is a one-to-one function, this implies that ( ab = a + 9b ). That seems like a key equation to work with.Let me write that down:[ ab = a + 9b ]I can rearrange this equation to maybe express one variable in terms of the other. Let's try to solve for ( a ):[ ab - a = 9b ][ a(b - 1) = 9b ][ a = frac{9b}{b - 1} ]Hmm, so ( a ) is expressed in terms of ( b ). Since ( a ) and ( b ) are positive real numbers, the denominator ( b - 1 ) must be positive, which means ( b > 1 ). That's an important point because it tells me the range of possible values for ( b ).Now, let me look at each statement one by one.**Statement A: The minimum value of ( ab ) is 36.**From the equation ( ab = a + 9b ), I can see that ( ab ) is expressed in terms of ( a ) and ( b ). Maybe I can use some inequality here to find the minimum value. The Arithmetic Mean-Geometric Mean (AM-GM) inequality often helps in such optimization problems.Let me consider the right-hand side ( a + 9b ). I can think of this as the sum of two terms: ( a ) and ( 9b ). Applying AM-GM to these two terms:[ frac{a + 9b}{2} geq sqrt{a cdot 9b} ][ frac{a + 9b}{2} geq sqrt{9ab} ][ frac{a + 9b}{2} geq 3sqrt{ab} ]Multiplying both sides by 2:[ a + 9b geq 6sqrt{ab} ]But from the original equation, ( ab = a + 9b ), so substituting:[ ab geq 6sqrt{ab} ]Let me let ( x = sqrt{ab} ), so ( x^2 = ab ). Then the inequality becomes:[ x^2 geq 6x ][ x^2 - 6x geq 0 ][ x(x - 6) geq 0 ]Since ( x = sqrt{ab} ) is positive, this inequality holds when ( x geq 6 ). Therefore, ( sqrt{ab} geq 6 ), so ( ab geq 36 ). That means the minimum value of ( ab ) is indeed 36. To check when equality holds, we need ( a = 9b ) (from the AM-GM equality condition). Let me substitute ( a = 9b ) into the equation ( ab = a + 9b ):[ 9b cdot b = 9b + 9b ][ 9b^2 = 18b ][ 9b^2 - 18b = 0 ][ 9b(b - 2) = 0 ]Since ( b > 1 ), ( b = 2 ). Then ( a = 9 times 2 = 18 ). So, ( a = 18 ) and ( b = 2 ) give ( ab = 36 ). Therefore, statement A is correct.**Statement B: The minimum value of ( frac{81}{a^2} + frac{1}{b^2} ) is ( frac{1}{2} ).**This seems a bit more complex. Let me see if I can express ( frac{81}{a^2} + frac{1}{b^2} ) in terms of a single variable using the relationship ( ab = a + 9b ).From earlier, I have ( a = frac{9b}{b - 1} ). Let me substitute this into ( frac{81}{a^2} + frac{1}{b^2} ):[ frac{81}{left( frac{9b}{b - 1} right)^2} + frac{1}{b^2} ]Simplify the first term:[ frac{81}{frac{81b^2}{(b - 1)^2}} = frac{81 cdot (b - 1)^2}{81b^2} = frac{(b - 1)^2}{b^2} ]So the expression becomes:[ frac{(b - 1)^2}{b^2} + frac{1}{b^2} = frac{(b - 1)^2 + 1}{b^2} ]Expanding ( (b - 1)^2 ):[ frac{b^2 - 2b + 1 + 1}{b^2} = frac{b^2 - 2b + 2}{b^2} ]Simplify:[ 1 - frac{2}{b} + frac{2}{b^2} ]Hmm, that doesn't immediately look like something I can minimize easily. Maybe I should consider another approach.Alternatively, since ( ab = a + 9b ), let me denote ( ab = k ). Then ( k = a + 9b ). From statement A, we know ( k geq 36 ).Let me express ( frac{81}{a^2} + frac{1}{b^2} ) in terms of ( k ). Since ( ab = k ), ( b = frac{k}{a} ). Substituting into the expression:[ frac{81}{a^2} + frac{1}{left( frac{k}{a} right)^2} = frac{81}{a^2} + frac{a^2}{k^2} ]Let me denote ( x = a^2 ), so the expression becomes:[ frac{81}{x} + frac{x}{k^2} ]This is of the form ( frac{m}{x} + frac{x}{n} ), which has a minimum value when ( frac{m}{x} = frac{x}{n} ), so ( x = sqrt{mn} ). Applying this here:Minimum occurs when ( frac{81}{x} = frac{x}{k^2} )[ x^2 = 81k^2 ][ x = 9k ]So, substituting back:[ frac{81}{x} + frac{x}{k^2} = frac{81}{9k} + frac{9k}{k^2} = frac{9}{k} + frac{9}{k} = frac{18}{k} ]Since ( k geq 36 ), the minimum value of ( frac{18}{k} ) is ( frac{18}{36} = frac{1}{2} ). Therefore, the minimum value of ( frac{81}{a^2} + frac{1}{b^2} ) is ( frac{1}{2} ), and it occurs when ( k = 36 ), which is when ( a = 18 ) and ( b = 2 ). So, statement B is correct.**Statement C: The minimum value of ( a + b ) is 16.**Again, using the relationship ( ab = a + 9b ), I need to find the minimum of ( a + b ). Let me express ( a + b ) in terms of ( b ) using ( a = frac{9b}{b - 1} ):[ a + b = frac{9b}{b - 1} + b = frac{9b + b(b - 1)}{b - 1} = frac{9b + b^2 - b}{b - 1} = frac{b^2 + 8b}{b - 1} ]Let me denote ( f(b) = frac{b^2 + 8b}{b - 1} ). I need to find the minimum of this function for ( b > 1 ).To find the minimum, I can take the derivative of ( f(b) ) with respect to ( b ) and set it to zero.First, compute the derivative ( f'(b) ):Using the quotient rule:[ f'(b) = frac{(2b + 8)(b - 1) - (b^2 + 8b)(1)}{(b - 1)^2} ]Simplify the numerator:[ (2b + 8)(b - 1) = 2b(b - 1) + 8(b - 1) = 2b^2 - 2b + 8b - 8 = 2b^2 + 6b - 8 ]Subtract ( (b^2 + 8b) ):[ 2b^2 + 6b - 8 - b^2 - 8b = b^2 - 2b - 8 ]So, ( f'(b) = frac{b^2 - 2b - 8}{(b - 1)^2} )Set ( f'(b) = 0 ):[ b^2 - 2b - 8 = 0 ]Solve the quadratic equation:[ b = frac{2 pm sqrt{4 + 32}}{2} = frac{2 pm sqrt{36}}{2} = frac{2 pm 6}{2} ]So, ( b = frac{8}{2} = 4 ) or ( b = frac{-4}{2} = -2 ). Since ( b > 1 ), we discard ( b = -2 ) and take ( b = 4 ).Now, compute ( a + b ) when ( b = 4 ):[ a = frac{9 times 4}{4 - 1} = frac{36}{3} = 12 ]So, ( a + b = 12 + 4 = 16 ).To confirm this is a minimum, let me check the second derivative or analyze the behavior around ( b = 4 ). Alternatively, since we found a critical point and the function tends to infinity as ( b ) approaches 1 from the right and as ( b ) approaches infinity, this critical point must be a minimum.Therefore, the minimum value of ( a + b ) is indeed 16. So, statement C is correct.**Statement D: The maximum value of ( frac{9a}{a + 1} + frac{b}{b + 1} ) is ( frac{100}{11} ).**This one seems trickier. Let me see how I can approach this. I need to find the maximum of ( frac{9a}{a + 1} + frac{b}{b + 1} ) given ( ab = a + 9b ).First, let me express ( a ) in terms of ( b ) again: ( a = frac{9b}{b - 1} ). Substitute this into the expression:[ frac{9 times frac{9b}{b - 1}}{frac{9b}{b - 1} + 1} + frac{b}{b + 1} ]Simplify the first term:Denominator: ( frac{9b}{b - 1} + 1 = frac{9b + (b - 1)}{b - 1} = frac{10b - 1}{b - 1} )So, the first term becomes:[ frac{9 times frac{9b}{b - 1}}{frac{10b - 1}{b - 1}} = frac{81b}{10b - 1} ]Therefore, the entire expression is:[ frac{81b}{10b - 1} + frac{b}{b + 1} ]Let me denote this as ( f(b) = frac{81b}{10b - 1} + frac{b}{b + 1} ). I need to find its maximum for ( b > 1 ).To find the maximum, I can take the derivative of ( f(b) ) with respect to ( b ) and set it to zero.First, compute the derivative ( f'(b) ):[ f'(b) = frac{81(10b - 1) - 81b(10)}{(10b - 1)^2} + frac{(1)(b + 1) - b(1)}{(b + 1)^2} ]Simplify each term:First term:[ frac{81(10b - 1) - 810b}{(10b - 1)^2} = frac{810b - 81 - 810b}{(10b - 1)^2} = frac{-81}{(10b - 1)^2} ]Second term:[ frac{(b + 1) - b}{(b + 1)^2} = frac{1}{(b + 1)^2} ]So, ( f'(b) = frac{-81}{(10b - 1)^2} + frac{1}{(b + 1)^2} )Set ( f'(b) = 0 ):[ frac{-81}{(10b - 1)^2} + frac{1}{(b + 1)^2} = 0 ][ frac{1}{(b + 1)^2} = frac{81}{(10b - 1)^2} ]Take square roots on both sides:[ frac{1}{b + 1} = frac{9}{10b - 1} ]Cross-multiplying:[ 10b - 1 = 9(b + 1) ][ 10b - 1 = 9b + 9 ][ 10b - 9b = 9 + 1 ][ b = 10 ]So, the critical point is at ( b = 10 ). Let me compute ( a ) when ( b = 10 ):[ a = frac{9 times 10}{10 - 1} = frac{90}{9} = 10 ]Now, compute the expression ( frac{9a}{a + 1} + frac{b}{b + 1} ) at ( a = 10 ) and ( b = 10 ):[ frac{9 times 10}{10 + 1} + frac{10}{10 + 1} = frac{90}{11} + frac{10}{11} = frac{100}{11} ]So, the expression reaches ( frac{100}{11} ) at ( a = 10 ) and ( b = 10 ). Now, I need to check if this is indeed a maximum.Looking at the behavior of ( f(b) ) as ( b ) approaches 1 from the right:- ( frac{81b}{10b - 1} ) approaches ( frac{81 times 1}{10 - 1} = 9 )- ( frac{b}{b + 1} ) approaches ( frac{1}{2} )So, ( f(b) ) approaches ( 9 + frac{1}{2} = 9.5 ), which is less than ( frac{100}{11} approx 9.09 ). Wait, that doesn't make sense because 9.5 is greater than 9.09. Hmm, maybe my intuition is off.Wait, actually, as ( b ) approaches 1 from the right, ( a = frac{9b}{b - 1} ) approaches negative infinity, but since ( a ) must be positive, ( b ) cannot approach 1 from the right. Wait, no, ( b ) must be greater than 1, so as ( b ) approaches 1 from the right, ( a ) approaches positive infinity. So, let me re-examine the limits.As ( b ) approaches 1 from the right:- ( a ) approaches infinity.- ( frac{9a}{a + 1} ) approaches 9.- ( frac{b}{b + 1} ) approaches ( frac{1}{2} ).So, ( f(b) ) approaches ( 9 + frac{1}{2} = 9.5 ).As ( b ) approaches infinity:- ( a = frac{9b}{b - 1} ) approaches 9.- ( frac{9a}{a + 1} ) approaches ( frac{81}{10} = 8.1 ).- ( frac{b}{b + 1} ) approaches 1.So, ( f(b) ) approaches ( 8.1 + 1 = 9.1 ).At ( b = 10 ), ( f(b) = frac{100}{11} approx 9.09 ).Wait, so as ( b ) increases beyond 10, ( f(b) ) approaches 9.1, which is slightly higher than 9.09. Hmm, that suggests that the maximum might actually be higher than ( frac{100}{11} ). But wait, when I computed the derivative, I found a critical point at ( b = 10 ), but the function seems to approach a higher value as ( b ) increases. That might mean that ( b = 10 ) is actually a minimum, not a maximum.Wait, let me double-check my derivative calculation. Earlier, I had:[ f'(b) = frac{-81}{(10b - 1)^2} + frac{1}{(b + 1)^2} ]Setting this equal to zero:[ frac{-81}{(10b - 1)^2} + frac{1}{(b + 1)^2} = 0 ][ frac{1}{(b + 1)^2} = frac{81}{(10b - 1)^2} ]Taking square roots:[ frac{1}{b + 1} = frac{9}{10b - 1} ]Cross-multiplying:[ 10b - 1 = 9(b + 1) ][ 10b - 1 = 9b + 9 ][ b = 10 ]So, the critical point is indeed at ( b = 10 ). To determine if it's a maximum or minimum, let me test values around ( b = 10 ).Take ( b = 9 ):Compute ( f'(9) ):[ f'(9) = frac{-81}{(90 - 1)^2} + frac{1}{(10)^2} = frac{-81}{841} + frac{1}{100} approx -0.096 + 0.01 = -0.086 ]Negative derivative, so function is decreasing at ( b = 9 ).Take ( b = 11 ):Compute ( f'(11) ):[ f'(11) = frac{-81}{(110 - 1)^2} + frac{1}{(12)^2} = frac{-81}{11881} + frac{1}{144} approx -0.0068 + 0.00694 approx 0.00014 ]Positive derivative, so function is increasing at ( b = 11 ).Therefore, at ( b = 10 ), the function changes from decreasing to increasing, which means ( b = 10 ) is a minimum, not a maximum. That contradicts my earlier conclusion. So, the critical point at ( b = 10 ) is actually a minimum.Wait, but earlier, when I plugged ( b = 10 ) into the expression, I got ( frac{100}{11} approx 9.09 ). However, as ( b ) approaches 1 from the right, the expression approaches 9.5, which is higher. As ( b ) approaches infinity, the expression approaches 9.1, which is also higher than 9.09. Therefore, the function ( f(b) ) has a minimum at ( b = 10 ) but no maximum because as ( b ) approaches 1 or infinity, the expression approaches higher values.But wait, when ( b ) approaches 1 from the right, ( a ) approaches infinity, but let's see what happens to ( frac{9a}{a + 1} ). As ( a ) approaches infinity, ( frac{9a}{a + 1} ) approaches 9. Similarly, ( frac{b}{b + 1} ) approaches ( frac{1}{2} ). So, the expression approaches ( 9 + frac{1}{2} = 9.5 ).Similarly, as ( b ) approaches infinity, ( a ) approaches 9, so ( frac{9a}{a + 1} ) approaches ( frac{81}{10} = 8.1 ), and ( frac{b}{b + 1} ) approaches 1, so the expression approaches ( 8.1 + 1 = 9.1 ).Therefore, the expression ( frac{9a}{a + 1} + frac{b}{b + 1} ) has a minimum value of ( frac{100}{11} ) at ( b = 10 ), but it doesn't have a maximum because it approaches 9.5 as ( b ) approaches 1. However, the problem states that ( a ) and ( b ) are positive real numbers, so ( b ) cannot be equal to 1, but it can get arbitrarily close to 1, making the expression approach 9.5.But the statement says the maximum value is ( frac{100}{11} ), which is approximately 9.09, but we just saw that the expression can get as high as 9.5. Therefore, the maximum value is actually 9.5, not ( frac{100}{11} ). So, statement D is incorrect.Wait, but let me double-check. Maybe I made a mistake in interpreting the behavior as ( b ) approaches 1. Let me compute ( f(b) ) when ( b ) is very close to 1, say ( b = 1.1 ):[ a = frac{9 times 1.1}{1.1 - 1} = frac{9.9}{0.1} = 99 ]Compute ( frac{9a}{a + 1} + frac{b}{b + 1} ):[ frac{9 times 99}{99 + 1} + frac{1.1}{1.1 + 1} = frac{891}{100} + frac{1.1}{2.1} approx 8.91 + 0.5238 approx 9.4338 ]Which is less than 9.5 but still higher than ( frac{100}{11} approx 9.09 ).Wait, so as ( b ) approaches 1, ( f(b) ) approaches 9.5, but when ( b = 1.1 ), it's about 9.43. So, maybe the maximum is actually 9.5, but it's not achieved because ( b ) can't be 1. So, the expression can get arbitrarily close to 9.5 but never actually reaches it. Therefore, the expression doesn't have a maximum value; it can get as close to 9.5 as desired. However, the problem states that ( a ) and ( b ) are positive real numbers, so technically, the expression doesn't have a maximum—it can approach 9.5 but never exceed it.But the statement says the maximum value is ( frac{100}{11} ), which is approximately 9.09. Since the expression can actually get higher than that, the statement is incorrect.Wait, but earlier, I found that at ( b = 10 ), the expression is ( frac{100}{11} approx 9.09 ), which is a minimum. So, the maximum is actually unbounded above 9.09, approaching 9.5. Therefore, the maximum value is not ( frac{100}{11} ); instead, that's the minimum. So, statement D is incorrect.Alternatively, maybe I made a mistake in interpreting the critical point. Let me try another approach. Maybe using substitution or another inequality.Given ( ab = a + 9b ), let me express ( a ) as ( a = frac{9b}{b - 1} ) as before. Then, the expression ( frac{9a}{a + 1} + frac{b}{b + 1} ) becomes:[ frac{9 times frac{9b}{b - 1}}{frac{9b}{b - 1} + 1} + frac{b}{b + 1} ]Simplify the first term:[ frac{frac{81b}{b - 1}}{frac{9b + (b - 1)}{b - 1}} = frac{81b}{10b - 1} ]So, the expression is:[ frac{81b}{10b - 1} + frac{b}{b + 1} ]Let me combine these two fractions:Find a common denominator, which is ( (10b - 1)(b + 1) ):[ frac{81b(b + 1) + b(10b - 1)}{(10b - 1)(b + 1)} ]Expand the numerator:[ 81b^2 + 81b + 10b^2 - b = 91b^2 + 80b ]So, the expression becomes:[ frac{91b^2 + 80b}{(10b - 1)(b + 1)} ]I need to find the maximum of this expression for ( b > 1 ).Alternatively, maybe I can express this as a function and analyze its behavior. Let me denote ( f(b) = frac{91b^2 + 80b}{(10b - 1)(b + 1)} ).To find the maximum, I can take the derivative and set it to zero, but this might get complicated. Alternatively, maybe I can perform polynomial division or simplify the expression.Let me try to simplify ( f(b) ):[ f(b) = frac{91b^2 + 80b}{10b^2 + 9b - 1} ]Let me perform polynomial long division:Divide ( 91b^2 + 80b ) by ( 10b^2 + 9b - 1 ).First term: ( frac{91b^2}{10b^2} = 9.1 ). Multiply the denominator by 9.1:[ 9.1 times (10b^2 + 9b - 1) = 91b^2 + 81.9b - 9.1 ]Subtract this from the numerator:[ (91b^2 + 80b) - (91b^2 + 81.9b - 9.1) = -1.9b + 9.1 ]So, ( f(b) = 9.1 + frac{-1.9b + 9.1}{10b^2 + 9b - 1} )This shows that as ( b ) approaches infinity, ( f(b) ) approaches 9.1. The term ( frac{-1.9b + 9.1}{10b^2 + 9b - 1} ) approaches zero. So, the expression approaches 9.1 from below because the numerator is negative for large ( b ).Wait, but earlier, I saw that as ( b ) approaches 1 from the right, ( f(b) ) approaches 9.5. So, the function ( f(b) ) starts near 9.5 when ( b ) is just above 1, decreases to a minimum at ( b = 10 ) (which is ( frac{100}{11} approx 9.09 )), and then increases again towards 9.1 as ( b ) approaches infinity. Therefore, the function has a minimum at ( b = 10 ) and approaches 9.5 and 9.1 as ( b ) approaches 1 and infinity, respectively.But since 9.5 is higher than 9.1, the maximum value the expression can approach is 9.5, but it never actually reaches it. Therefore, the expression doesn't have a maximum value; it can get arbitrarily close to 9.5. However, the problem states that the maximum value is ( frac{100}{11} approx 9.09 ), which is actually the minimum. Therefore, statement D is incorrect.Alternatively, maybe I made a mistake in interpreting the critical point. Let me check the second derivative to confirm the nature of the critical point at ( b = 10 ).Compute the second derivative ( f''(b) ):From earlier, ( f'(b) = frac{-81}{(10b - 1)^2} + frac{1}{(b + 1)^2} )Differentiate again:[ f''(b) = frac{1620}{(10b - 1)^3} - frac{2}{(b + 1)^3} ]At ( b = 10 ):[ f''(10) = frac{1620}{(99)^3} - frac{2}{(11)^3} ]Compute each term:[ frac{1620}{970299} approx 0.00167 ][ frac{2}{1331} approx 0.00150 ]So, ( f''(10) approx 0.00167 - 0.00150 = 0.00017 ), which is positive. Therefore, the function is concave upwards at ( b = 10 ), confirming that it's a minimum.Thus, the expression ( frac{9a}{a + 1} + frac{b}{b + 1} ) has a minimum value of ( frac{100}{11} ) at ( a = 10 ) and ( b = 10 ), but it doesn't have a maximum value because it can approach 9.5 as ( b ) approaches 1. Therefore, statement D is incorrect.**Summary:**- Statement A: Correct- Statement B: Correct- Statement C: Correct- Statement D: IncorrectSo, the correct conclusions are A, B, and C.

💡Okay, so I have this problem where I need to find the value of the function ( f(a, b, c) = frac{a^4}{a^3 + b^2 + c^2} + frac{b^4}{b^3 + a^2 + c^2} + frac{c^4}{c^3 + b^2 + a^2} ) given that ( a, b, c ) are positive real numbers satisfying ( a + b + c = 1 ). First, I think it might be helpful to consider the case where ( a = b = c ). Since ( a + b + c = 1 ), each of them would be ( frac{1}{3} ). Let me compute ( f ) at this point.So, substituting ( a = b = c = frac{1}{3} ) into ( f(a, b, c) ):Each term becomes ( frac{(frac{1}{3})^4}{(frac{1}{3})^3 + (frac{1}{3})^2 + (frac{1}{3})^2} ).Calculating the numerator: ( (frac{1}{3})^4 = frac{1}{81} ).Calculating the denominator: ( (frac{1}{3})^3 + 2 times (frac{1}{3})^2 = frac{1}{27} + 2 times frac{1}{9} = frac{1}{27} + frac{2}{9} = frac{1}{27} + frac{6}{27} = frac{7}{27} ).So each term is ( frac{1/81}{7/27} = frac{1}{81} times frac{27}{7} = frac{1}{21} ).Since there are three such terms, the total is ( 3 times frac{1}{21} = frac{1}{7} ).So, ( f(frac{1}{3}, frac{1}{3}, frac{1}{3}) = frac{1}{7} ).Hmm, interesting. Maybe this is the minimum value of ( f(a, b, c) )? I should check if this is indeed the case.To see if ( f(a, b, c) geq frac{1}{7} ), I might need to use some inequality techniques. Let me think about using the Cauchy-Schwarz inequality or perhaps the Titu's lemma, which is a specific case of Cauchy-Schwarz.Titu's lemma states that ( frac{x_1^2}{y_1} + frac{x_2^2}{y_2} + dots + frac{x_n^2}{y_n} geq frac{(x_1 + x_2 + dots + x_n)^2}{y_1 + y_2 + dots + y_n} ).In our case, each term is of the form ( frac{a^4}{a^3 + b^2 + c^2} ). Let me see if I can apply Titu's lemma here.Let me denote ( x_i = a^2 ) and ( y_i = a^3 + b^2 + c^2 ) for each term. Then, by Titu's lemma:( f(a, b, c) geq frac{(a^2 + b^2 + c^2)^2}{(a^3 + b^3 + c^3) + 2(a^2 + b^2 + c^2)} ).Okay, so I need to express this in terms of known quantities. Since ( a + b + c = 1 ), I can relate ( a^2 + b^2 + c^2 ) and ( a^3 + b^3 + c^3 ) to this.First, ( a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) = 1 - 2(ab + bc + ca) ).Let me denote ( S = ab + bc + ca ). So, ( a^2 + b^2 + c^2 = 1 - 2S ).Similarly, ( a^3 + b^3 + c^3 ) can be expressed using the identity:( a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ca) + 3abc ).Substituting ( a + b + c = 1 ), we get:( a^3 + b^3 + c^3 = 1 - 3S + 3abc ).Let me denote ( P = abc ). So, ( a^3 + b^3 + c^3 = 1 - 3S + 3P ).Now, substituting back into the inequality from Titu's lemma:( f(a, b, c) geq frac{(1 - 2S)^2}{(1 - 3S + 3P) + 2(1 - 2S)} ).Simplify the denominator:( (1 - 3S + 3P) + 2(1 - 2S) = 1 - 3S + 3P + 2 - 4S = 3 - 7S + 3P ).So, the inequality becomes:( f(a, b, c) geq frac{(1 - 2S)^2}{3 - 7S + 3P} ).Now, I need to relate ( S ) and ( P ). I know that for positive real numbers ( a, b, c ) with ( a + b + c = 1 ), the maximum of ( S ) occurs when two variables are equal and the third is zero, but since all variables are positive, the maximum ( S ) is less than ( frac{1}{3} ).Wait, actually, the maximum of ( S ) occurs when ( a = b = c = frac{1}{3} ), giving ( S = 3 times frac{1}{3} times frac{1}{3} = frac{1}{3} ). So, ( S leq frac{1}{3} ).Similarly, the minimum of ( S ) is greater than zero since all variables are positive.But I also know that for fixed ( a + b + c ), the product ( abc ) is maximized when ( a = b = c ). So, ( P leq left( frac{1}{3} right)^3 = frac{1}{27} ).But I'm not sure how to use this yet. Maybe I can find a relationship between ( S ) and ( P ).Alternatively, perhaps I can use the AM-GM inequality on the denominator.Wait, let me think differently. Maybe I can find a lower bound for ( f(a, b, c) ) by considering the function when variables are unequal.Suppose one variable is close to 1 and the other two are close to 0. Let me test this case.Let ( a = 1 - epsilon ), ( b = epsilon ), ( c = 0 ), where ( epsilon ) is a small positive number approaching 0.But wait, ( c ) must be positive, so let me take ( c = epsilon ) as well, so ( a = 1 - 2epsilon ), ( b = epsilon ), ( c = epsilon ).Compute ( f(a, b, c) ):First term: ( frac{a^4}{a^3 + b^2 + c^2} = frac{(1 - 2epsilon)^4}{(1 - 2epsilon)^3 + 2epsilon^2} ).As ( epsilon ) approaches 0, this term approaches ( frac{1}{1} = 1 ).Second term: ( frac{b^4}{b^3 + a^2 + c^2} = frac{epsilon^4}{epsilon^3 + (1 - 2epsilon)^2 + epsilon^2} ).Simplify denominator: ( epsilon^3 + 1 - 4epsilon + 4epsilon^2 + epsilon^2 = 1 - 4epsilon + 5epsilon^2 + epsilon^3 ).So, the term is approximately ( frac{epsilon^4}{1} approx 0 ).Similarly, the third term is the same as the second term, so it also approaches 0.Thus, as ( epsilon ) approaches 0, ( f(a, b, c) ) approaches 1. But wait, this contradicts the earlier result where ( f(a, b, c) = frac{1}{7} ) when ( a = b = c ). So, perhaps the function can take values both above and below ( frac{1}{7} )?Wait, no, when ( a ) approaches 1 and ( b, c ) approach 0, ( f(a, b, c) ) approaches 1, which is greater than ( frac{1}{7} ). So, maybe ( frac{1}{7} ) is the minimum value.But earlier, when I set ( a = b = c = frac{1}{3} ), I got ( f(a, b, c) = frac{1}{7} ). So, perhaps ( frac{1}{7} ) is indeed the minimum, and the function can go up to 1.But the problem is to "find" ( f(a, b, c) ). It doesn't specify whether it's asking for the minimum, maximum, or something else. Maybe it's just to compute it, but given the constraints, it's likely asking for the minimum.Wait, the problem says "find" without specifying, but given the context, it's probably asking for the minimum value.So, going back, I have:( f(a, b, c) geq frac{(1 - 2S)^2}{3 - 7S + 3P} ).I need to show that this is at least ( frac{1}{7} ).So, let's set up the inequality:( frac{(1 - 2S)^2}{3 - 7S + 3P} geq frac{1}{7} ).Cross-multiplying:( 7(1 - 2S)^2 geq 3 - 7S + 3P ).Expanding the left side:( 7(1 - 4S + 4S^2) geq 3 - 7S + 3P ).Which simplifies to:( 7 - 28S + 28S^2 geq 3 - 7S + 3P ).Bringing all terms to the left:( 7 - 28S + 28S^2 - 3 + 7S - 3P geq 0 ).Simplify:( 4 - 21S + 28S^2 - 3P geq 0 ).So, ( 28S^2 - 21S + 4 geq 3P ).Now, I need to relate ( P ) to ( S ). I know that for positive real numbers ( a, b, c ) with ( a + b + c = 1 ), the maximum of ( P ) is ( frac{1}{27} ) when ( a = b = c = frac{1}{3} ). Also, from the AM-GM inequality, ( P leq left( frac{a + b + c}{3} right)^3 = frac{1}{27} ).But I need a lower bound for ( P ). Wait, actually, I need an upper bound for ( 3P ), because in the inequality ( 28S^2 - 21S + 4 geq 3P ), the right side is ( 3P ), so I need to show that ( 28S^2 - 21S + 4 ) is at least as big as ( 3P ).But since ( P leq frac{1}{27} ), ( 3P leq frac{1}{9} ). So, if I can show that ( 28S^2 - 21S + 4 geq frac{1}{9} ), then the inequality would hold.Let me compute ( 28S^2 - 21S + 4 - frac{1}{9} = 28S^2 - 21S + frac{35}{9} ).I need to show that ( 28S^2 - 21S + frac{35}{9} geq 0 ).This is a quadratic in ( S ). Let me compute its discriminant:Discriminant ( D = (-21)^2 - 4 times 28 times frac{35}{9} = 441 - frac{3920}{9} ).Compute ( 441 = frac{3969}{9} ), so ( D = frac{3969 - 3920}{9} = frac{49}{9} ).So, the roots are:( S = frac{21 pm sqrt{49/9}}{2 times 28} = frac{21 pm 7/3}{56} ).Calculating:First root: ( frac{21 + 7/3}{56} = frac{63/3 + 7/3}{56} = frac{70/3}{56} = frac{70}{168} = frac{5}{12} ).Second root: ( frac{21 - 7/3}{56} = frac{63/3 - 7/3}{56} = frac{56/3}{56} = frac{1}{3} ).So, the quadratic ( 28S^2 - 21S + frac{35}{9} ) is positive outside the interval ( [frac{1}{3}, frac{5}{12}] ).But since ( S leq frac{1}{3} ) (as established earlier), the quadratic is positive for ( S leq frac{1}{3} ).Therefore, ( 28S^2 - 21S + frac{35}{9} geq 0 ) for ( S leq frac{1}{3} ), which is always true.Thus, ( 28S^2 - 21S + 4 geq 3P ), which implies ( f(a, b, c) geq frac{1}{7} ).Therefore, the minimum value of ( f(a, b, c) ) is ( frac{1}{7} ), achieved when ( a = b = c = frac{1}{3} ).

💡First, identify the number of blue marbles that Jason has, which is 44.Next, determine the number of blue marbles that Tom has, which is 24.To find out how many more blue marbles Jason has compared to Tom, subtract the number of blue marbles Tom has from the number of blue marbles Jason has.Perform the subtraction: 44 minus 24 equals 20.Therefore, Jason has 20 more blue marbles than Tom.

💡First, I need to determine the least common multiple (LCM) of the days each librarian works: 5, 8, 10, and 14.I'll start by factoring each number into its prime factors:- 5 is a prime number, so its prime factorization is simply 5.- 8 can be factored into (2^3).- 10 can be factored into (2 times 5).- 14 can be factored into (2 times 7).Next, I'll identify the highest power of each prime number present in these factorizations:- The highest power of 2 is (2^3) (from 8).- The highest power of 5 is 5 (from 5 and 10).- The highest power of 7 is 7 (from 14).Finally, I'll calculate the LCM by multiplying these highest powers together:[text{LCM} = 2^3 times 5 times 7 = 8 times 5 times 7 = 40 times 7 = 280]Therefore, all four librarians will next work together in 280 days.

💡Okay, so I need to figure out how much Spongebob earned in a day from selling burgers and fries. Let's see, he sold 30 burgers at 2 each and 12 large fries at 1.5 each. Hmm, okay, so I think I need to calculate the total money he made from the burgers and then from the fries, and then add them together to get the total earnings.First, let's tackle the burgers. He sold 30 burgers, and each burger costs 2. So, to find out how much he made from the burgers alone, I should multiply the number of burgers by the price per burger. That would be 30 times 2. Let me write that down: 30 * 2 = 60. Okay, so he made 60 from selling burgers.Now, onto the fries. He sold 12 large fries, and each one costs 1.5. So, similarly, I need to multiply the number of fries by the price per fry to find out how much he made from fries. That would be 12 times 1.5. Let me calculate that: 12 * 1.5 = 18. So, he made 18 from selling fries.Now, to find out his total earnings for the day, I need to add the money he made from burgers and the money he made from fries. That would be 60 plus 18. Let me add those together: 60 + 18 = 78.Wait, does that make sense? Let me double-check my calculations to make sure I didn't make any mistakes. For the burgers: 30 burgers at 2 each. 30 times 2 is indeed 60. For the fries: 12 fries at 1.5 each. 12 times 1.5 is 18. Adding them together: 60 plus 18 is 78. Yeah, that seems correct.Is there any other cost involved that I might have missed? The problem doesn't mention any costs or expenses, just the earnings from selling the burgers and fries. So, I think I'm safe to say that Spongebob earned 78 for the day.Just to make sure, let me think about it another way. If he sold 30 burgers at 2 each, that's like 30 times 2, which is 60. And 12 fries at 1.5 each is 12 times 1.5, which is 18. Adding those together gives 78. Yep, that matches what I got before.Alright, I think I've got it right. Spongebob earned 78 for the day.