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Suppose $f’(a) = M$ where $M > 0$. Prove that there is a $\delta>0$ such that if $0<|x-a|<\delta$, then $\frac{f(x)-f(a)}{x-a} > M/2$.

I am probably making this problem harder than it is. I would appreciate a push in the right direction. The solution isn't coming to me. Thanks!

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    If the derivative is $M$, then as $x \to a$ the difference quotient approaches $M$, and so must eventually be larger than $M/2$ when $x$ is near enough to $a$. – 2012-11-08
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    Consider using [LaTeX](http://meta.math.stackexchange.com/questions/107/faq-for-math-stackexchange/117#117). – 2012-11-08
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    The way it is phrased now, this is not true. Consider $f(x)=0 \forall x \in \mathbb{R}$, then $f'(a)=0 \forall a \in \mathbb{R}$ but the difference quotient does not exist, so certainly it cannot be positive. – 2012-11-08

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