(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.
Accepted Solution
A:
[tex]f(x,y)=x^4+y^4-4xy+1[/tex]has critical points wherever the partial derivatives vanish:[tex]f_x=4x^3-4y=0\implies x^3=y[/tex][tex]f_y=4y^3-4x=0\implies y^3=x[/tex]Then[tex]x^3=y\implies x^9=x\implies x(x^8-1)=0\implies x=0\text{ or }x=\pm1[/tex]If [tex]x=0[/tex], then [tex]y=0[/tex]; critical point at (0, 0)If [tex]x=1[/tex], then [tex]y=1[/tex]; critical point at (1, 1)If [tex]x=-1[/tex], then [tex]y=-1[/tex]; critical point at (-1, -1)[tex]f(x,y)[/tex] has Hessian matrix[tex]H(x,y)=\begin{bmatrix}12x^2&-4\\-4&12y^2\end{bmatrix}[/tex]with determinant[tex]\det H(x,y)=144x^2y^2-16[/tex]At (0, 0), the Hessian determinant is -16, which indicates a saddle point.At (1, 1), the determinant is 128, and [tex]f_{xx}(1,1)=12[/tex], which indicates a local minimum.At (-1, -1), the determinant is again 128, and [tex]f_{xx}(-1,-1)=12[/tex], which indicates another local minimum.