Question

Find all the local maxima, local minima and saddle points of the function.
f(x,y) = x e^y

Answer 1

fx(x,y)=e^y fy(x,y)=xe^y


e^y=0 x.e^y=0

y=1 x=0

(0,1)


fxx(x,y)=0,fxy(x,y)=xe^y,fxy(x,y)=e^y

D=-e^2 <0 Therfore,Saddle point.

Answer 2

Please check

Answer 3

Wouldn't the question be something like this?
f(x,y) = xe^(-2x^2 - 2y^2)

Answer 4

Both correct !
(0,1) is a local minima.
D = fxx fyy - fxy^2 = -e^2 < 0
so (0,1) is a saddle point.

Answer 5

How is it a local minima @math&ing001

Answer 6

You already found that it's a local minima/maxima. To check which one it is, plug in any random pair and compare it with f(0,1).