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Math Help - Prove a function is constant.

  1. #1
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    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.

    Please help, thank you.
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  2. #2
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    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→∞.
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