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

it's equivalent, so we can say that its equal.

let's call the denominator x and the numerador is 18 less then denominator, x - 18.

multiply

2.x = 5.(x - 18)

2x = 5x - 90

90 = 5x - 2x

90 = 3x

x = 90/3

x = 30

so: