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.