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Thread: I think my algebra died

  1. #1
    Member billym's Avatar
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    Red face I think my algebra died

    Why am I getting stuck on these questions?

    Find and classify the stationary points:

    $\displaystyle f(x,y)=(x-y)^3+x^2y^2$

    $\displaystyle f_x=3(x-y)^2+2xy^2$

    $\displaystyle f_y=-3(x-y)^2+2x^2y$

    I forget how to find the roots. What to I need to revise? Any links?
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  2. #2
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    Quote Originally Posted by billym View Post
    Why am I getting stuck on these questions?

    Find and classify the stationary points:

    $\displaystyle f(x,y)=(x-y)^3+x^2y^2$

    $\displaystyle f_x=3(x-y)^2+2xy^2$

    $\displaystyle f_y=-3(x-y)^2+2x^2y$

    I forget how to find the roots. What to I need to revise? Any links?
    It is not trivial and requires quite some tricky algebra:

    $\displaystyle f_x=0=f_y \Longrightarrow 3(x-y)^2+2xy^2=-3(x-y)^2+2x^2y$$\displaystyle \Longrightarrow 6(x-y)^2-2xy(x-y)=0 \Longrightarrow$

    $\displaystyle \Longrightarrow 3(x-y)\left[2(x-y)-xy\right]=0$ $\displaystyle \Longrightarrow x=y\,,\,\,or \,\,2(x-y)=xy\Longleftrightarrow x-y=\frac{xy}{2}$

    If x = y then plugging into $\displaystyle f_x$ we get $\displaystyle x=y=0$, else we plug $\displaystyle x-y=\frac{xy}{2}\,\,in\,\,f_x\,,\,\,f_y$ and:

    $\displaystyle f_x=3\left(\frac{xy}{2}\right)^2+2xy^2 \Longrightarrow xy^2 \left(\frac{3}{4}x+2\right)=0$, and again $\displaystyle x = y = 0$ or else $\displaystyle x=-\frac{8}{3}$,
    and doing the same with the other partial derivative we get that also $\displaystyle y=-\frac{8}{3}$

    Nervertheless these last two values for x,y don't work, as you can check at once, so we're left with only...

    Tonio
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  3. #3
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    $\displaystyle
    \Longrightarrow 6(x-y)^2-2xy(x-y)=0 \Longrightarrow
    $
    $\displaystyle
    \Longrightarrow 3(x-y)\left[2(x-y)-xy\right]=0
    $
    There is a mistake there, it is supposed to be


    $\displaystyle
    \Longrightarrow 2(x-y)\left[3(x-y)-xy\right]=0
    $
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