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Westkost [7]
3 years ago
13

(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, l

ocal maximum or saddle point.
Mathematics
1 answer:
lara31 [8.8K]3 years ago
4 0

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.
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