# Find critical points and test for relative extrema.

Not what you're looking for? Search our solutions OR ask your own Custom question.

List the critical points for which the second partials test fails.

f(x,y)=x^3+y^3-6x^2+9y^2+12x+27y+19

Â© BrainMass Inc. brainmass.com November 29, 2021, 11:54 pm ad1c9bdddfhttps://brainmass.com/math/calculus-and-analysis/find-critical-points-test-relative-extrema-9020

#### Solution Preview

First we find the critical points of the function f(x,y):

f1(x,y) is the first derivative with respect to x and f2(x,y) is the first derivative with respect to y. So:

f1(x,y) := 3*x^2-12*x+12

f2(x,y) := 3*y^2+18*y+27

We have to solve the system of f1=0 and f2=0 and find all the common (x,y) roots of both. As you can see f1 has no y element and f2 has no x component, so you can solve them separately.

f1=0 -> x=2, x=2 (repeated)

f2=0 -> y=-3, y=-3 (repeated)

...

#### Solution Summary

Critical points are found and a function is tested for extrema. The second partial test fails are listed for a function.

$2.49