A local maximum (or minimum) of a function is a point inside the domain in which our function takes a value greater than its value on its neighbors.
A point q at which f has non-zero directional derivative in any direction in which we can move both forward and back, cannot be a maximum or minimum, since moving in that direction from q forward and back will cause f to increase one way and decrease in the other.
The basic condition for an interior "extremum" at point q of a differentiable function f is then that f have zero derivative in every direction that you are allowed to move from q while obeying the conditions on your problem.
For a function of one variable this is just the condition that f ' = 0 at x = q, which is to say that q is a critical point for f.
To find whether f has a maximum or minimum at a critical point you must look to the quadratic approximation (or if necessary to the first higher approximation at which f deviates from flatness) to f. If its second derivative is positive then, like x2, f has a minimum at q, and if it is negative f has a maximum.
You should always check whether any local maximum or minimum that you find
is the "global" maximum or minimum of f. That global extreme point
(or any such points) can occur on a boundary, or at a different local extremum
from the first one you find.
If f is a function of several variables then strange things can go on even in the quadratic approximation, and q being a critical point does not imply that it is a maximum or minimum even when the quadratic approximation is far from flat. As noted in Chapter 11, q could easily be a saddle point and you have to check for that as indicated in that chapter.
To solve such problems you look for critical points of f by setting all its derivatives to zero, and solving the resulting equations. Then you must check whether you have a maximum, minimum or saddle point.
Here we consider what happens when you are in 2 or 3 dimensions but are looking for an extreme point of a function F among points that lie on some surface or curve.