Function with non-negative second derivative and 2 zeros (incomplete)
Dependencies: Unspecified
Let $f: \mathbb{R} \mapsto \mathbb{R}$ be a function such that
- $f$ is differentiable in $[a, b]$.
- $\forall x \in [a, b], f''(x) \ge 0$.
- $f(a) = f(b) = 0$.
Then $\forall x \in [a, b], f(x) \le 0$.
Proof idea
(Rigorous proof needed)
Assume that $\exists t \in (a, b)$ such that $f(t) > 0$. Then by mean value theorem, $\exists x_1 \in (0, t)$ such that $f'(x_1) > 0$ and $\exists x_2 \in (t, 1)$ such that $f'(x_2) < 0$. But this contradicts the fact that $f''(x) \ge 0$. Therefore, such a $t$ doesn't exist.
Dependency for:
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- Number of transitive dependencies: 0