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The problem asks about the function \( g(x) = x^4 \), and has two parts:

### Part (a) asks for:
- \( g'(x) \), the first derivative of \( g(x) \), and
- \( g''(x) \), the second derivative of \( g(x) \).

Let's calculate these derivatives.

1. **First derivative \( g'(x) \):**
   - \( g(x) = x^4 \)
   - By differentiating \( x^4 \) with respect to \( x \), we get:
   \[
   g'(x) = 4x^3
   \]

2. **Second derivative \( g''(x) \):**
   - Differentiating \( g'(x) = 4x^3 \) with respect to \( x \), we get:
   \[
   g''(x) = 12x^2
   \]

### Part (b) asks to deduce the convexity of the function \( g \).

Convexity is determined by the second derivative \( g''(x) \):
- If \( g''(x) > 0 \), the function is **convex**.
- If \( g''(x) < 0 \), the function is **concave**.

Since \( g''(x) = 12x^2 \), which is always non-negative (and only zero at \( x = 0 \)), we can conclude that:
- \( g(x) \) is **convex** for all \( x \in \mathbb{R} \), and it is strictly convex except at \( x = 0 \), where it is flat (neither convex nor concave).

Would you like further details on any of these steps?

### Related Questions:
1. What is the geometric interpretation of a function's convexity?
2. How do the signs of the first and second derivatives affect the shape of a graph?
3. Can you give examples of concave and convex functions other than polynomials?
4. How does inflection occur in terms of second derivatives?
5. What is the relationship between higher-order derivatives and the function’s behavior?

### Tip:
For polynomial functions like \( x^4 \), the degree of the terms influences the growth rate, which in turn impacts their convexity or concavity.

g est la fonction définie sur ℝ par: g(x) = x^4. 
a) Déterminer mentalement g'(x) et g''(x). 
b) En déduire la convexité de la fonction g.

Learn how to calculate the first and second derivatives of the function g(x) = x^4 and deduce its convexity. This problem explores key calculus concepts, including differentiation and the second derivative test for convexity. Suitable for students in Grades 11-12.

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