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Thread: 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 $\displaystyle f(x,y) = e^{-xy}(x^2 + y^2)$

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

    I just took both partials and found:

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

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

    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.

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

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

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

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

    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 $\displaystyle f_{xx}f_{yy} - (f_{xy})^2$, 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 $\displaystyle f(x,y) = e^{-xy}(x^2 + y^2)$

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

    I just took both partials and found:

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

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

    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 $\displaystyle e^u$ is never zero for real $\displaystyle u$, so:

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

    implies that:

    $\displaystyle -x^2y-y^3+2x=0$

    and similarly:

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

    implies that:

    $\displaystyle -x^3-xy^2+2y=0$

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

    CB
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