微積分極大極小值題目

Question

Find the local maximum and minimum values and saddle points

of f(x.y)=x^2 +y^2 +1/((x^2)(y^2)) x.y不等於0

Answer 1

最佳解答

Grad(f)=(∂f/∂x, ∂f/∂y)=(2x-2/(x^3 y^2), 2y-2/(x^2 y^3))=(0,0)

then x^2=y^2=1,

so, (x,y)=(1,1),(1,-1),(-1,1),(-1,-1)

matrix of the second derivatives D=

[ 2+ 6/(x^4 y^2) 4/(x^3y^3) ]

[ 4/(x^3 y^3) 2+6/(x^2 y^4) ]

at the critical points, D=

[ 8 4 ] or [ 8 -4 ]

[ 4 8 ] [-4 8 ]

both are positive definite ( 8>0, det(D)>0)

so f(1, 1)=f(1,-1)=...=f(-1,-1)= 3 are minimum values (no max. ...)