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© BrainMass Inc. brainmass.com February 4, 2021, 6:29 pm ad1c9bdddf
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)
Critical points are found and a function is tested for extrema. The second partial test fails are listed for a function.