f(x,y)=3x2−3xy+2y2−9x−3y−11

Question

find all local maxima, local minima, and saddle points

 

Answer 1

f(x,y)=3x²-3xy+2y²-9x-3y-11.

∂f/∂x=6x-3y-9=0 at an extremum (1)

∂f/∂y=-3x+4y-3=0 at an extremum (2)

∂²f/∂x²=6, ∂²f/∂y²=4, ∂f/∂x∂y=-3

Solve these simultaneous equations:

(1)+2(2): 5y-15=0, 5y=15, y=3.

So 6x-9-9=0, 6x=18, x=3.

Now we need to test what type of extremum (3,3) is.

D=(∂²f/∂x²)(∂²f/∂y²)-(∂f/∂x∂y)² at (3,3):

D=24-9=15.

Since D and ∂²f/∂x² are both >0, there is a local minimum at (3,3).

f(3,3)=27-27+18-27-9-11=-29.

There is a local minimum at (3,3,-29).