Let f : (0, ∞ ) →R be a differentiable function such that f'(x2) = 1 -x3 for all x > 0 and f(1) = 0.Then f(4) equalsa)-47/5b)-47/10c)-16/5d)-8/5Correct answer is option 'A'. Can you explain this answer?

Question

Let f : (0, &#8734; ) &#8594;R be a differentiable function such that f'(x2) = 1 -x3 for all x &#62; 0 and f(1) = 0.Then f(4) equalsa)-47/5b)-47/10c)-16/5d)-8/5Correct answer is option 'A'. Can you explain this answer?

Answer

Put √x in given eqn , then integrate it , you will get f(x) = x-2x^(5/2)/5+c. put f(1)= 0 to get c . Then easily you can find f(4)

Answer

Given Information

The function f : (0,∞) → R is differentiable.
f(x^2) = 1 - x^3 for all x > 0
f(1) = 0

Solution

We need to find the value of f(4).

Step 1: Differentiating the given equation

Let's differentiate the given equation f(x^2) = 1 - x^3 with respect to x.

Using the chain rule, we have:
2x * f'(x^2) = -3x^2

Simplifying the equation, we get:
f'(x^2) = -3x / (2x)

Simplifying further, we have:
f'(x^2) = -3/2

Step 2: Finding the derivative of f(x)

To find the derivative of f(x), we substitute x^2 with x in the derivative we obtained in Step 1.

f'(x) = -3 / (2√x)

Step 3: Integrating the derivative to find f(x)

Integrating f'(x) with respect to x, we obtain f(x):

f(x) = -3 / 2 * ∫ (1 / √x) dx

Integrating, we get:
f(x) = -3 / 2 * 2√x + C

Simplifying further, we have:
f(x) = -3√x + C

Step 4: Finding the value of C

Using the given condition f(1) = 0, we substitute x = 1 into the equation f(x) = -3√x + C.

0 = -3√1 + C

0 = -3 + C

C = 3

Step 5: Finding f(4)

Substituting x = 4 into the equation f(x) = -3√x + C, we have:

f(4) = -3√4 + 3

f(4) = -3 * 2 + 3

f(4) = -6 + 3

f(4) = -3

Thus, the value of f(4) is -3, which is not listed as one of the given options. Therefore, the given answer options seem to be incorrect.

Please note that option 'A' (-47/5) is not the correct answer based on the given information.

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