6

6. Find an classify all the critical points of

Solution: Note first z has the same value for

(x,y), (x, -y), (x, -y), (-x, -y)

so we only have to analyze what is going on in the first quadrant

Going to the first partials,

Taking the ratio,

Since x⁴y² = 1 it follows that x⁶ = 1, and x = 1, y = 1.

Going to the second partials

So, D = 8*8-16 >0, A > 0 and, consequently, there is a minimum at (1,1) (and also at (1, -1), (-1,1), (-1,-1))