In Evans' book "Partial Differential Equations", on page 348, it is mentioned that the outer normal derivative at the maximum point (if it exists) $\displaystyle \frac{\partial u}{\partial \nu}(x_0) \ge 0$ is obvious. Why is it true?
Is it also true that for any continuous functions on an open bounded domain, the outer normal derivative (assume it exists) at the maximum point is non-negative? I recall in Calculus 1-3, I always have either the first derivative to be 0 or not existing at the maximum point. How can I get a strictly positive inequality?
Think of the one-dimensional case. If $f$ is a differentiable function on a closed interval $[a,b]$, you'll have a local maximum at the endpoint $b$ if $f'(b) > 0$, or at the endpoint $a$ if $f'(a) < 0$.