Question

(a)5%)Let fx, y) = x^4 + y^4 - 4xy + 1. and classify each critical point Find all critical points of fx,y) as a local minimum, local maximum or saddle point.

Answer 1

`f(x,y)=x^4+y^4-4xy+1`

has critical points wherever the partial derivatives vanish:

`f_x=4x^3-4y=0\implies x^3=y`

`f_y=4y^3-4x=0\implies y^3=x`

Then

`x^3=y\implies x^9=x\implies x(x^8-1)=0\implies x=0\text{ or }x=\pm1`

• If
  `x=0`, then
  `y=0`; critical point at (0, 0)
• If
  `x=1`, then
  `y=1`; critical point at (1, 1)
• If
  `x=-1`, then
  `y=-1`; critical point at (-1, -1)

`f(x,y)` has Hessian matrix

`H(x,y)=\begin{bmatrix}12x^2&-4\\-4&12y^2\end{bmatrix}`

with determinant

`\det H(x,y)=144x^2y^2-16`

• At (0, 0), the Hessian determinant is -16, which indicates a saddle point.
• At (1, 1), the determinant is 128, and
  `f_(xx)(1,1)=12`, which indicates a local minimum.
• At (-1, -1), the determinant is again 128, and
  `f_(xx)(-1,-1)=12`, which indicates another local minimum.