Classify the critical points of the function f t y x2 y2 e

Classify the critical points of the function f (t. y) = (x^2 + y^2) e^y^2-x^2 No local minimum, no local maximum, saddle points at (0.0), (1.0) and (- 1.0). local minimum 0 at (0.0). local maximum 2 at (1.1). saddle points at (1.0) and (-1.0). No local minimum, local maximum 8 at (2.2). saddle points at (1.0) and (-1 0).

Solution

f(x,y)=(x^2+y^2)e^(y2-x2)

Fx= e^(Y^2-X^2)(2X)+ e^(Y^2-X^2)(X^2+Y^2)(-2X)

=   e^(Y^2-X^2)(2X)[1-X^2-Y^2]

Fy= e^(Y^2-X^2)[2Y+(X^2+Y^2)(2Y)]

= 2Y e^(Y^2-X^2)[1+X^2+Y^2]

From here we can find Fxx, Fxy and Fyy

from here Fxx=2e^(Y^2-X^2)[1-X^2-Y^2]

Fxy=2e^(2X)(1-Y^2-X^2)

Fyy=2e^(-2Y)(1-Y^2-X^2)

from here there are (0,0)

on substituting this we have no local minimum,no local maximum, and saddle points at (0,0),(1,0) and (-1,0)