# Prove a function is constant.

• Nov 16th 2007, 08:26 PM
Suppose that $f:R \mapsto R$ is twice differentiable everywhere; $f(x) \leq 0$ and $f''(x) \geq 0 \ \ \ \ \ \forall x$. Prove that $f:R \mapsto R$ is constant.
Suppose that $f'(x_0)>0$ for some point x_0. Since f"(x)≥0, it follows that f' is an increasing function, and so f'(x)≥f'(x_0) for all x≥x_0. That means that f(x)→+∞ as x→+∞, contradicting the fact that f(x) is never positive.