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If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =?
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If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin...
Local Maximum Value of Function f(x)


Given Function:

f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2)


Local Maximum Value:

The local maximum value of the given function is K/e.


Solution


Step 1: Calculating First Derivative

To find the local maximum value, we need to calculate the first derivative of the given function.


f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x)

Using the chain rule:

f'(x) = (sin^2 x) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * (-2cos x / (2sin^2 x))

f'(x) = -((sqrt(3e))/(sin x)) * (sin^2 x) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * cos x / sin x

f'(x) = -((sqrt(3e))^2/(2sin^3 x)) * (sin^2 x) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * cos x


Step 2: Calculating Second Derivative

Now, we need to calculate the second derivative of the given function.


f'(x) = -((sqrt(3e))^2/(2sin^3 x)) * (sin^2 x) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * cos x

Using the product rule:

f''(x) = -((sqrt(3e))^2/(2sin^3 x)) * (sin^2 x) * [(sin^2 x - 1) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 2) * (-2cos x / (2sin^2 x)) + ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * (-2cos x / (2sin^2 x))] - ((sqrt(3e))^2/(2sin^3 x)) * ln((sqrt(3e))/(2sin x)) * ((sqrt(3e))/(2sin x)) ^ (sin^2 x - 1) * 2sin x / (2sin^3 x) * (sin^2 x)

f''(x) = ((sqrt(3e))^4/(4sin^5 x)) * [(sin^2 x
Community Answer
If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin...
Local Maximum Value of Function


We are given the function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) and we need to find the local maximum value of the function.


Solution


Step 1: Finding the Derivative of the Function


To find the local maximum value of the function, we need to find the derivative of the function.


f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x)

ln(f(x)) = (sin^2(x)) * ln(sqrt(3e)/(2sin(x)))

ln(f(x)) = (sin^2(x)) * (ln(sqrt(3e)) - ln(2sin(x)))

ln(f(x)) = (sin^2(x)) * (ln(sqrt(3e)) - ln(2) - ln(sin(x)))

ln(f(x)) = (sin^2(x)) * ln(sqrt(3e)/2) - (sin^2(x)) * ln(sin(x))


Now, we can find the derivative of ln(f(x)):


ln(f(x))' = 2sin(x) * ln(sqrt(3e)/2) - sin^2(x)/sin(x)


We can simplify this expression further:


ln(f(x))' = 2sin(x) * ln(sqrt(3e)/2) - sin(x)


Step 2: Finding the Critical Points of the Function


To find the critical points of the function, we need to set the derivative equal to 0:


ln(f(x))' = 2sin(x) * ln(sqrt(3e)/2) - sin(x) = 0


2sin(x) * ln(sqrt(3e)/2) = sin(x)


ln(sqrt(3e)/2) = 1/2


sqrt(3e)/2 = e^(1/2)


sqrt(3e) = 2e^(1/2)


3e = 4e


This equation has no solution in the interval (0, pi/2).


Step 3: Finding the Second Derivative of the Function


To determine whether the critical point is a local maximum or a local minimum, we need to find the second derivative of the function:


ln(f(x))'' = 2ln(sqrt(3e)/2) - cos(x)


ln(f(x))'' = 2ln(sqrt(3e)/2) - cos(x)


ln(f(x))'' = 2ln(e^(1/2)) - cos(x)


ln(f(x))'' = 1 - cos(x)


Step 4: Evaluating the Second Derivative at the Critical Point


Since there is no critical point, we cannot evaluate the second
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If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =?
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If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =?.
Solutions for If local maximum value of function f(x) = ((sqrt(3e))/(2sin x)) ^ (sin^2 x); ,x in (0, pi/2) is K/e then (K/e) ^ 8 (k ^ theta)/(e ^ 5) K ^ theta =? in English & in Hindi are available as part of our courses for JEE. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
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