# Thread: Prove a function is constant.

1. ## Prove a function is constant.

Problem:

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.

I'm trying to prove this thing by showing f'(x) = 0 for all x. Now, I know f' is either monotonically increasing or constant. But I have problem using the f <= 0 for my proof.

2. 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.

Similarly, if f'(x) is ever negative then you can show that f(x)→+∞ as x→–∞.