Reason for a lower gradient value

Question

We know that the gradient will be $0$ at maxima or minima. Let's assume we measure the gradient at two points and that at the first point the norm of the gradient is higher than at the second point. Can we assume that the second point is always nearer to a local/global maxima or minima?

Answer 1

Generally, no, because the function could do just about anything outside of those two points. For example:

1. It could reach 0 but have an inflection point - e.g. $y = x^3$ which has no local extrema;

2. It could be approaching an asymptote - e.g. $y = \frac{x}{x^2 + 1}$ where it reaches a maximum at $(1, \frac{1}{2})$, then drops off and smooths down to a horizontal asymptote of $y = 0$ as $x \rightarrow \infty$ so that from $x = \sqrt{3}$ onwards you'll always see the gradient getting smaller the further you get from the actual critical point.

3. It could just stop existing (there's no requirement that the function is even defined on any particular domain).