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Let $f: (a, b) \to \mathbb{R}$ and $f$ differentiable in $x_0 \in (a, b)$ and $f'(x_0) \ \neq 0$

I want to prove that $g(x) = (f(x) - f(x_0))(x - x_0)$ has in $x_0$ severe extremum.

Looks like $f$ is monotonous and we can prove in this case. But what can we say about $g(x)$ to prove this fact?

Sarvesh Ravichandran Iyer
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    $g$ is not defined at $x_0$. – Fred Jan 19 '21 at 14:13
  • The one thing you know about $f$ is that for $\epsilon > 0, \exists \delta > 0$ such that in $(x_0 - \delta, x_0 + \delta)$, $$f'(x_0) -\epsilon < \dfrac{f(x) - f(x_0)}{x-x_0} < f'(x_0) +\epsilon$$. – Paul Sinclair Jan 19 '21 at 20:42

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