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Math Help - Did I do this right? (Local max/min/partial derivatives)

  1. #1
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    Did I do this right? (Local max/min/partial derivatives)

    Let f(x,y) = e^{-xy}(x^2 + y^2)

    1) Find polynomial equations for the critical points of f.

    I just took both partials and found:

    f_{x}=-ye^{-xy}(x^2 + y^2) + 2x(e^{-xy}) = e^{-xy}(-x^2y-y^3+2x)

    f_{x}=-xe^{-xy}(x^2 + y^2) + 2y(e^{-xy}) = e^{-xy}(-x^3-xy^2+2y)

    It's asking for polynomial equations, so do I just factor the exponent e out and write the equations left?

    2) Find all the second order partial derivatives of f.

    f_{xx}=e^{-xy}(y^4+x^2y^2-4xy+2)

    f_{yy}=e^{-xy}(x^4+x^2y^2-4xy+2)

    f_{xy}=e^{-xy}(x^3y+xy^3-3x^2-3y^2)

    As it should be, I found f_{xy} = f_{yx}.

    3) It can be shown that the critical points of f are (0,0), (1,1) and (-1,-1). Classify these points.

    By plugging the points into f_{xx}f_{yy} - (f_{xy})^2, I got that (0,0) was a minimum, (1,1) was a maximum, and (-1,-1) was a saddle point.

    You probably are wondering what I need help with; I really just wanted reassurance on the first part and wanted to make sure I was getting the concept right. I have a test coming up soon, and I wanted to make sure these review problems were getting done correctly so that I was prepared.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by wilcofan3 View Post
    Let f(x,y) = e^{-xy}(x^2 + y^2)

    1) Find polynomial equations for the critical points of f.

    I just took both partials and found:

    f_{x}=-ye^{-xy}(x^2 + y^2) + 2x(e^{-xy}) = e^{-xy}(-x^2y-y^3+2x)

    f_{y}=-xe^{-xy}(x^2 + y^2) + 2y(e^{-xy}) = e^{-xy}(-x^3-xy^2+2y)

    It's asking for polynomial equations, so do I just factor the exponent e out and write the equations left?
    It requires you to observe that e^u is never zero for real u, so:

    f_{x}=-ye^{-xy}(x^2 + y^2) + 2x(e^{-xy}) = e^{-xy}(-x^2y-y^3+2x)=0

    implies that:

    -x^2y-y^3+2x=0

    and similarly:

    f_{y}=-xe^{-xy}(x^2 + y^2) + 2y(e^{-xy}) = e^{-xy}(-x^3-xy^2+2y)=0

    implies that:

    -x^3-xy^2+2y=0

    Which is a pair of simultaneous polynomial equations for the critical points.

    CB
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