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# Find critical points and test for relative extrema.

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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

https://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.

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