Question

Given the equation z = x^2 + y^3 + xy, find the critical points of the equation and identify wether they are mins maxes or neither.

I got the points (0,0,0) neither, (-1/12, 1/6, -1/432) neither. Is this right?

Answer 1

z=f(x,y)
f(x,y) = x^2+y^3+xy
Then find the first partial derivatives (fx) and (fy)
fx= 2x+y=0 ,fy=3y^2+x=0
The critical points are (0,0) and (-1/12,1/6)
Using the second derivative test (0,0) is a saddle point
While (-1/12,1/6) is a minimum