Find the absolute maximum and absolute minimum of the function f(x, y)=2x3+y4f(x, y)=2x3+y4 on the region {(x, y)|x2+y2≤64}{(x, y)|x2+y2≤64} ignore unneeded answer blanks, and list points lexicographically.

Check where the first-order partial derivatives vanish to find any critical points within the given region:

The Hessian for this function is

with , so unfortunately the second partial derivative test fails. However, if we take we see that for different values of ; if we take we see takes on both positive and negative values. This indicates (0, 0) is neither the site of an extremum nor a saddle point.

Now check for points along the boundary. We can parameterize the boundary by

with . This turns into a univariate function :

At these critical points, we get

We only care about 3 of these results.

So to recap, we found that attains

s

step-by-step explanation:

. + λrh() e 1 1 2 4 4 4 4 4 3 4 4 4 4 4 + th−1(− s(x)2 dx ω + λrh−1() e 2 1 2 4 4

step-by-step explanation: