Question

Find the local max & min values & saddle points of the function.

f(x,y)=x*y*e^(-x^2-y^2)

Answer 1

For my partial derivatives

Fx=e^(-x^2-y^2) * (y-2x^2y) & Fy=e^(-x^2-y^2) * (x-2xy^2)

Answer 2

Now after i set them 0 i dont know how to get the critical points

Answer 3

Hint: e^x > 0 for all x

Answer 4

the critical points are (0,0) (\[(\pm1/\sqrt{2},\pm1/\sqrt{2}) (+1/\sqrt{2},-1/\sqrt{2}) and (-1/\sqrt{2}, +1/\sqrt{2})\])

Answer 5

so when e^(x) > 0 the function goes to 1/sqrt(2)

Answer 6

Now that you've got all the critical points, just perform the Second Derivatives Test, the result should drop out.