Finding the critical points?

Question

f(x,y)= e^(-x^2)(y^2+1)

i got fx= =2xe^(-x^2)(y^2+1)
fy=2ye^(-x^2)

my problem is solving the equations and getting the critical points. Also, how can we get y in terms of x to solve the equation???

help much appreciated

Answer 1

Your partial derivatives are correct.

The equations you have to solve are

2x e^(-x²)(y²+1) = 0 and
2y e^(-x²) = 0

2x e^(-x²)(y²+1) = 0 requires
x = 0 or e^(-x²) = 0 or y²+1 = 0

e^(-x²) is never zero, and y²+1 cannot be zero for any real y.
Therefore, the only solution to ∂f/∂x = 0 is x=0

Similarly,

2y e^(-x²) = 0 requires
y = 0 or e^(-x²) = 0

By the same reasoning as above, the only solution for ∂f/∂y = 0 is y=0

Therefore, the only critical point is (0, 0)