Question

Find the critical points of f(x,y)=x^2+y^4+2xy and classify them as local maximum, local minimum or saddle point

Answer 1

Answer:

The unique critical point is (x,y)=(−2,0) and it is a local minimum (and the local minimum value of the function is f(−2,0)=4-8+0=-4#.

Explanation:

The first-order partial derivatives of z=f(x,y)=x2+4x+y2 are ∂z∂x=2x+4 and ∂z∂y=2y.

Setting both of these equal to zero results in a system of equations whose unique solution is clearly (x,y)=(−2,0), so this is the unique critical point of f.

The second-order partials are ∂2z∂x2=2, ∂2z∂y2=2, and ∂2z∂x∂y=∂2z∂y∂x=0

This makes the discriminant for the (multivariable) Second Derivative Test equal to

D=∂2z∂x2⋅∂2z∂y2−(∂2z∂x∂y)

Answer 2

Answer:

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