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Suppose $$ F(a)=0, F(b)=0, a$F''(x)<0$ for $a.

Do we have $F(x) > 0$ for $a?

Can you tell me an existing theorem that I can use in a paper to obtain the conclusion? I think there should be one, but I can't remember.

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    If $F(x) < 0$ for some $a < x < b$, then you can use the mean value theorem three times to show that $F''(X) > 0$ somewhere.2012-12-11
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    @arkeet I mean, can you tell me an existing theorem that I can use in a paper to obtain the conclusion? I think there should be one, but I can't remember.2012-12-11

3 Answers 3