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If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q?
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If maximum value of integral 0 to 1 (f(x))3 under the condition integr...
Solution:
Given:
- Maximum value of integral 0 to 1 (f(x))3 is to be found
- under the condition integral 0 to 1 f(x)dx is 0
- f(x) lies between -1 to 1, including both
Approach:
- We need to find the maximum value of integral 0 to 1 (f(x))^3 under the given condition.
- We can use the Lagrange multiplier method to solve the problem.
- We need to maximize the function F(f) = integral 0 to 1 (f(x))^3 + λ integral 0 to 1 f(x)dx
- Here, λ is the Lagrange multiplier.
- We also have the condition integral 0 to 1 f(x)dx = 0
- Using the Lagrange multiplier method, we can write the following equations:
- dF(f)/df = 3(f(x))^2 + λ = 0
- dF(f)/dλ = integral 0 to 1 f(x)dx = 0
- Solving the above equations, we get:
- λ = -3(f(x))^2
- integral 0 to 1 f(x)dx = 0
- Substituting λ in the first equation, we get:
- 3(f(x))^2 = -3(f(x))^3
- (f(x))^2(1 + f(x)) = 0
- So, f(x) can be either 0 or -1.
- However, we also have the condition that f(x) lies between -1 and 1, including both.
- Therefore, f(x) can only be 0.
- The maximum value of the integral 0 to 1 (f(x))^3 is 0.
- Hence, p/q = 0/1 = 0.
Community Answer
If maximum value of integral 0 to 1 (f(x))3 under the condition integr...
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If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q?
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If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q?.
If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If maximum value of integral 0 to 1 (f(x))3 under the condition integral 0 to 1 f(x)dx is 0 and f(x) lies between -1 to 1, including both is p/q, then find p+q?.
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