Find the partial derivative of (x，y) with respect to x
Find the partial derivative of (x，y) with respect to y
Find the critical points of the function:
Equate these two partial derivatives to zero.
from（2）
Substitute x=-y in equation(1) then we have
At y=0,
from（2）
At y= -1
from（2）
X=1
Hence, the critical points of the function f are（0，0）and（1，-1）
Find the second partial derivatives of
Hence
Now partially differentiate the function with respect to y
Therefore,
Now partially differentiate the function with respect to x
Therefore
Find D(x,y)
At the critical point (0,0)
And
Clearly,D>0 and
Hence, the function is local minimum at（0，0）
The value of f at the point（0，0）is,
Hence, the critical point at（0，0）gives the local minimum value of f(0,0)=0
At the critical point（1，-1）
Clearly,D<0,
Therefore, the function has a saddle point at（-1，1）
The first order partial derivatives are calculated as,
For critical points set and ,then and
Solving these two equations as follows:，
Rewrite first equation as, and
So, the critical points of the function are
（0，0）,(0,2),(-2,0),(-2,2)
The second order partial derivatives are calculated as follows:
At critical point（0，0）, the values are calculated as,
