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Thread: Finding CP using Partial Derivatives

  1. #1
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    Finding CP using Partial Derivatives

    The equation is this
    B = e ^(xy/2)

    I have successfully found all 6 first and second order partial derivatives but need help finding the CP [/FONT]
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  2. #2
    Senior Member Pinkk's Avatar
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    I assume you mean critical point? If that's what you're talking about, then $\displaystyle (a,b)$ is a critical if $\displaystyle (a,b)$ is in the domain of $\displaystyle f(x,y)$ and:

    1) $\displaystyle \frac{\partial f}{\partial x}(a,b)=\frac{\partial f}{\partial y}(a,b)=0$

    OR

    2) one or both of $\displaystyle \frac{\partial f}{\partial x}(a,b)$ and $\displaystyle \frac{\partial f}{\partial y}(a,b)$ are undefined.

    In the case of $\displaystyle f(x,y)=e^{\frac{xy}{2}}$:

    $\displaystyle \frac{\partial f}{\partial x}=\frac{ye^{\frac{xy}{2}}}{2}$
    $\displaystyle \frac{\partial f}{\partial y}=\frac{xe^{\frac{xy}{2}}}{2}$

    Knowing this, $\displaystyle (0,0)$ must be a critical point because $\displaystyle \frac{\partial f}{\partial x}=0$ when $\displaystyle y=0$. Similarly, $\displaystyle \frac{\partial f}{\partial y}=0$ when $\displaystyle x=0$. Since $\displaystyle e^{\frac{xy}{2}}$ can never equal $\displaystyle 0$, the only critical point is $\displaystyle (0,0)$.
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