Question

Let f be a differentiable function on [alpha and beta ] and x =[alpha and beta]. Show that, if f '(x) =0 and f''(x) > 0, then f must have a local maximum at x.

Answer 1

Suppose $f$ has a local maximum at $x_0 ∈ (\alpha, \beta)$ . For small (enough) h, $f(x_0 + h) ≤ f(x0).$

If $h > 0$ then

$\dfrac{f(x_0 + h) − f(x_0)}{ h} ≤ 0.$

Similarly, if $h < 0$ , then

$\dfrac{f(x_0 + h) − f(x_0)}{ h} ≥ 0.$

By elementary properties of the limit, it follows that $f'(x_0) = 0.$