Problem:
Suppose thatis twice differentiable everywhere;
and
. Prove that
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.
Please help, thank you.
Suppose thatfor 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→–∞.
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