Home  18.013A  Chapter 14 


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 nonzero 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 $f\text{'}=0$ 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 ${x}^{2},f$ 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.
