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I am trying to prove the second derivative test. First, I need to prove the following lemma. Let $I$ be an open interval. Suppose that the function $f:I\to \mathbb{R}$ is continuous and that at the point $c$ in $I$, $f(c)> 0$. Prove that there is a $\delta > 0$ such that $f(x)> 0$ if $|x - c|< \delta$.

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    $\begingroup$ Your lemma does not appear to be connected to second derivative test. How are you going to use it? $\endgroup$ – Paramanand Singh Jan 11 '18 at 0:04
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Hint: what does continuity at $c$ mean? It means that you can always control how close the images of points near $c$ are from the image of $c$ just by taking points close enough to $c$. In particular, if you only consider points near a sufficiently small neighborhood of $c$, you can guarantee that their images are going to be close enough to $f(c)$ as to continue being strictly positive.

Now restate this in terms of epsilon-deltas and you will get your lemma.

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I have the answer. Just use f(c)/2 as the positive epsilon in the epsilon-delta criterion for continuity at a point.

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