Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ Justify that $f \in \mathcal{C}^1(\mathbb{R})$ and that $f$ is convex. Give the set of its minimizers.
Consider the function
$$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$
Justify that $f \in \mathcal{C}^1(\mathbb{R})$ and that $f$ is convex. Give the set of its minimizers.