Problem:
Let G: (0, ∞) → R be a differentiable function such that f'(x^2) = 1 - x^3 for all x > 0 and f(1) = 0. Determine the value of f(4).

Solution:

To find the value of f(4), we need to find an expression for f(x) and then substitute x = 4 into that expression.

Step 1: Finding f(x)

We are given that f'(x^2) = 1 - x^3 for all x > 0. To find f(x), we need to undo the derivative by integrating both sides of the equation with respect to x:

∫f'(x^2) dx = ∫(1 - x^3) dx

Step 2: Evaluating the integrals

Integrating the left side with respect to x gives us:

∫f'(x^2) dx = f(x^2) + C

where C is the constant of integration.

Integrating the right side with respect to x gives us:

∫(1 - x^3) dx = x - (x^4 / 4) + D

where D is another constant of integration.

Therefore, our equation becomes:

f(x^2) + C = x - (x^4 / 4) + D

Step 3: Determining the constants

We know that f(1) = 0, so we can substitute x = 1 into our equation to find the value of the constant C:

f(1^2) + C = 1 - (1^4 / 4) + D
C = 1/4 - D

Step 4: Simplifying the equation

Substituting C = 1/4 - D back into our equation, we get:

f(x^2) + (1/4 - D) = x - (x^4 / 4) + D

Simplifying further, we have:

f(x^2) = x - (x^4 / 4) + (D - 1/4)

Step 5: Substituting x = 4

Now we can substitute x = 4 into our equation to find the value of f(4):

f((4)^2) = 4 - (4^4 / 4) + (D - 1/4)

Simplifying further, we have:

f(16) = 4 - 64/4 + (D - 1/4)

f(16) = 4 - 16 + (D - 1/4)

f(16) = -12 + (D - 1/4)

Since we are not given any information about the value of D, we cannot determine the exact value of f(16). However, we can express f(16) in terms of D as:

f(16) = D - 49/4

Therefore, the value of f(4) is dependent on the value of D. Without any further information about D, we cannot determine the exact value of f(4).