Ask: Does a bounded function $f$ on $\mathbb R$, which is continuous and twice differentiable on $\mathbb R$, has zero for its second derivative, i.e. there exists a point $x_0 \in \mathbb R$ such that $$f''(x_0)=0$$ My idea is considering the contrapositive for the statement above. Let say $$f''(x_0)>0$$ , or equivalently the function $f$ is strictly convex. (Another case: Consider the function $-f$, which is strictly concave and having $-f''(x_0)<0$.) Then the function $f$ supposes to be strictly increasing, so $f$ is not bounded on $\mathbb R$.
Then it follows the conclusion is true.
I am doubted whether my proof is correct or not, and am interested in finding a relevant example, or a counterexample for the conclusion.