Let $f:(0,\infty)\rightarrow R$ be a differentiable function, such that $f^{\prime}(x^{2})=1-x^{3}$ for all $x0$ and $f(1)=0,$ then $f(4)=$

Question

Let $f:(0,\infty)\rightarrow R$ be a differentiable function, such that $f^{\prime}(x^{2})=1-x^{3}$ for all $x>0$ and $f(1)=0,$ then $f(4)=$

Accepted Answer

To solve the problem, we need to find the value of $ f(4) $ given the conditions of the function $ f $.

### Step-by-Step Explanation:

1. **Given Information:**
   - $ f:(0,\infty)\rightarrow \mathbb{R} $ is a differentiable function.
   - The derivative of $ f $ at $ x^2 $ is given by $ f^{\prime}(x^{2})=1-x^{3} $.
   - $ f(1) = 0 $.

2. **Objective:**
   - Find $ f(4) $.

3. **Understanding the Derivative:**
   - We have $ f^{\prime}(x^{2})=1-x^{3} $.
   - To find $ f(x) $, we need to express this in terms of $ x $.

4. **Substitution and Integration:**
   - Let $ u = x^2 $, then $ du = 2x \, dx $ or $ dx = \frac{du}{2x} $.
   - The derivative $ f^{\prime}(u) = 1 - x^3 $.
   - Since $ u = x^2 $, $ x = \sqrt{u} $, and thus $ x^3 = (\sqrt{u})^3 = u^{3/2} $.
   - Therefore, $ f^{\prime}(u) = 1 - u^{3/2} $.

5. **Integrating to Find $ f(u) $:**
   - Integrate $ f^{\prime}(u) = 1 - u^{3/2} $ with respect to $ u $:
     $$
     f(u) = \int (1 - u^{3/2}) \, du = u - \frac{2}{5}u^{5/2} + C
     $$

6. **Applying Initial Condition:**
   - We know $ f(1) = 0 $.
   - Substitute $ u = 1 $ into the integrated function:
     $$
     0 = 1 - \frac{2}{5}(1)^{5/2} + C \implies 0 = 1 - \frac{2}{5} + C
     $$
     $$
     C = \frac{2}{5} - 1 = -\frac{3}{5}
     $$

7. **Finding $ f(4) $:**
   - Now, substitute $ u = 4 $ into the function:
     $$
     f(4) = 4 - \frac{2}{5}(4)^{5/2} - \frac{3}{5}
     $$
   - Calculate $ (4)^{5/2} = (2^2)^{5/2} = 2^5 = 32 $.
   - Substitute back:
     $$
     f(4) = 4 - \frac{2}{5} \times 32 - \frac{3}{5}
     $$
     $$
     f(4) = 4 - \frac{64}{5} - \frac{3}{5}
     $$
     $$
     f(4) = 4 - \frac{67}{5}
     $$
     $$
     f(4) = \frac{20}{5} - \frac{67}{5} = \frac{-47}{5}
     $$

8. **Conclusion:**
   - The value of $ f(4) $ is $\frac{-47}{5}$, which corresponds to Option 3.

### Explanation of Incorrect Options:

- **Option 1: $\frac{-5}{8}$**
  - This value does not match the calculated result of $\frac{-47}{5}$ and arises from incorrect integration or arithmetic errors.
  
- **Option 2: $\frac{-8}{5}$**
  - This value is incorrect as it does not align with the derived expression for $ f(4) $.

- **Option 4: $\frac{-5}{47}$**
  - This value is incorrect and results from a misunderstanding of the integration process or arithmetic.

Thus, the correct answer is Option 3: $\frac{-47}{5}$.