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The exercise asks about the function \( f \) defined on \( \mathbb{R} \) by \( f''(x) = -2x^3 \).

The questions are:

1. **a)** Determine \( f'(x) \) and \( f(x) \) for all real \( x \).
2. **b)** Deduce the convexity of the function \( f \).

### Let's tackle this step by step:

### a) Determining \( f'(x) \) and \( f(x) \)

Given:
\[
f''(x) = -2x^3
\]
To find \( f'(x) \), integrate \( f''(x) \):
\[
f'(x) = \int -2x^3 \, dx = -\frac{2x^4}{4} + C_1 = -\frac{x^4}{2} + C_1
\]
Next, to find \( f(x) \), integrate \( f'(x) \):
\[
f(x) = \int \left(-\frac{x^4}{2} + C_1\right) dx = -\frac{x^5}{10} + C_1x + C_2
\]

Thus, the general forms are:
\[
f'(x) = -\frac{x^4}{2} + C_1
\]
\[
f(x) = -\frac{x^5}{10} + C_1x + C_2
\]

### b) Deduce the convexity of \( f \)

The convexity of a function depends on the sign of its second derivative \( f''(x) \). We already know:
\[
f''(x) = -2x^3
\]
- For \( x > 0 \), \( f''(x) < 0 \), meaning the function is concave.
- For \( x < 0 \), \( f''(x) > 0 \), meaning the function is convex.

Thus, the function is **convex** for \( x < 0 \) and **concave** for \( x > 0 \).

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Would you like more details on any part of the solution? Here are five related questions you might consider:

1. How does integrating a function twice give us the original function?
2. How do constants of integration affect the general solution of a differential equation?
3. What are the conditions for a function to be concave or convex?
4. How can we apply convexity and concavity in optimization problems?
5. What does the inflection point represent in terms of convexity and concavity?

**Tip:** Convexity and concavity are useful for determining the shape of a function and whether it has local minima or maxima.

f''(x) = -2x^3. a) Determine, for all real x, f'(x) and f(x). b) Deduce the convexity of the function f.

Solve the function f for f''(x) = -2x^3 by integrating twice to find f'(x) and f(x), and deduce the convexity of the function. This problem covers key calculus concepts such as differentiation, integration, and convexity.

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