For a function of one variable , you look for
local maxima and minima at * critical points* ---
points where the derivative is zero. You do
something similar to find maxima and minima for functions of two
variables.

A point is a * local max* if
for all points "near" . If you want to rule out "ties", then you
have to require instead that for all
points "near" . In this case, is a * strict local maximum*.

Similarly, a point is a * local
min* if for all points "near" . Again, to rule out "ties", you'd require
instead that for all points "near" . In this case, is a * strict local minimum*.

* Proposition.* Suppose f is defined on an open
set containing and is differentiable at . If is a local max or a local
min, then

* Proof.* If is a local max, then it is a local max of , a function of one variable x. Then by the result on local maxima from
single variable calculus. A similar argument shows that .

A critical point which is neither a local max nor a local min is a
* saddle*.

Thus, to find local maxima or minima, locate the *
critical points* by solving these equations simultaneously:

* Example.* Use graphs to describe the critical
points of , , , and .

has a local minimum at .

has a local maximum at .

has a saddle point at .

has a line of critical points along . They are local minima, but not strict local minima.

Once you've located the critical points, you must determine whether they are maxes, mins, or saddles. We have an analog of the Second Derivative Test for functions of one variable.

* Theorem.* Suppose f is differentiable on an
open set containing a critical point . Suppose the second partial derivatives are
continuous.

Define

Then:

(a) If , then is a saddle (neither a max nor a min).

(b) If and , then is a min.

(c) If and , then is a max.

(d) If , the test fails. (This doesn't mean the point is neither a max nor a min --- it means there is no conclusion.)

* Remarks.* In (b) and (c), you can use
" " instead of " ", because if the two must have the same sign.

To see that gives no conclusion, consider . You can check that is a critical point and that at . However, is a minimum. For , but if , then . Hence, every point gives a larger value for f than .

* Proof.* (Sketch) I'll sketch the argument for
this test in the case where .

There is a * Taylor expansion* for functions of
two variables:

Here is a point on the segment from to .

Since is a critical point, , and the series becomes

To save writing, let

The series can then be written

As long as h and k are small, I can assume that the second derivatives (and hence ) have the same signs at and . I'll consider the case where . Note that A and C must be both positive or both negative. For if one is positive and one is negative, then is negative, and , contrary to our assumption. So suppose that A and C are both positive. Then I can write

Since and , the last term is positive. I have

That is, for small h and k. This says that points close to give bigger f's, so must be a min.

Note that if , then is negative (because the stuff in the brackets is positive). Then

This says that points close to give smaller f's, so must be a max.

The argument that yields a saddle is more involved, so I'll omit it.

* Example.* Locate and classify the critical
points of

First, compute the partials:

Find the critical points:

The critical point is .

Here's a picture of the surface. You can see the min pretty clearly.

* Example.* Locate and classify the critical
points of

First, compute the partials:

Find the critical points:

Multiply by 4, multiply by 3, then add the equations:

Then

The critical point is .

* Example.* Locate and classify the critical
points of

First, compute the partials:

Set the first partials equal to zero and do a little simplification:

It's okay to divide out common factors *which are numbers*,
but *you should avoid dividing by something with a variable in
it* --- unless you're certain the expression cannot be zero.

I'll solve the equations simultaneously using a *
solution tree*. Start with one of the equations --- in this case,
it doesn't matter which one.

Every time I have an equation , I get two cases --- and --- and the tree splits into two branches. (In this case, I had a three-way split, but one branch closed up because there were no solutions.)

I continue working down a branch until I've solved for x and y. Notice that you bring in each of the original equations just once. This will help you avoid going in circles.

Finally, *you should not combine values from different
branches*. For example, I can't put together with to get .

Now test the critical points:

Using the table, it's easy to remember --- it's the second column times the third, minus the
square of the fourth. Notice that means the point is *either* a max or
a min. To decide which it is, look at . (This may seem asymmetric, but in fact,
if then and have the same
sign.)

Here's a picture of the surface. I've deformed it a bit to exaggerate the critical points.

* Example.* Locate and classify the critical
points of

First, compute the partials:

Set the first partials equal to zero:

Solve simultaneously:

Test the critical points:

Here's a picture of the surface, deformed to exaggerate the critical points:

* Example.* Locate and classify the critical
points of

First, compute the partials:

Set the first partials equal to zero:

Solve simultaneously:

Test the critical points:

Here's a picture of the surface, again deformed to exaggerate the critical points:

* Example.* Locate and classify the critical
points of

Compute the partial derivatives:

Set the first partials equal to zero:

Solve simultaneously:

Test the critical points:

Here's a picture of the surface, again deformed to exaggerate the critical points:

Copyright 2018 by Bruce Ikenaga