6 7 points each For the function fx y x3 6xy 8y3 a Find a

6. (7 points each) For the function f(x y) = x^3 - 6xy + 8y^3 a) Find all critical Points of f(x,y) b) Calculate the function D(x, y) C) Use D(x,y)to find any maximum local minimum and saddle points.

Solution

Given:

f(x)=x^3-6xy+8y^2

First differentiate the equation wrt to x and y:
           df/dx = 3x^2-6y
           df/dy = -6x+24y^2
Set both equations equal to 0 and solve the system.

This gives you the critical points:
     0=x^2-2y
     0=-x+3y^2
so critical points are (0,0) and (1,1/2).
To decide whether we have a maximum/minimum or saddle point, we need to investigate the second derivatives,

i.e. we must consider the Hessian matrix of f:
           H=<<6x,-6>|<-6,48y>>.
Compute the determinant of H.

          det H=288xy-36.
Now plug the solutions in:
          (0,0) -> -36<0 --> saddle point
(1,1/2) -> 108 >0 & the first entry df^2/(dxdy)|(1,1/2)=6*1=6 > 0 --> local minimum