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Thread: Critical point question

  1. #1
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    Critical point question

    hi everyone

    having problems finding critical points for this two equation to fined the maximum,minimum or saddle points.

    a) $\displaystyle x^4+y^4-4xy+1$
    $\displaystyle f_x=4x^3-4y$, $\displaystyle f_y=4y^3-4x$

    b) f(x,y)=$\displaystyle x^4-2x^2y+2y$
    $\displaystyle f_x=4x^3-4xy$,$\displaystyle f_y=2-2x^2$

    need help to find the critical points for this two equations,really hope someone can help me. really appreciate all your help & support.
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  2. #2
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    How far have you come in your attempts? Where do critical points for a given function occur?
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  3. #3
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    thank you for replying.

    a) $\displaystyle x^3-y=0$,$\displaystyle y^3-x=0$

    $\displaystyle x^3=y$
    $\displaystyle y^3=x$
    im stuck here, unable to find the critical point
    b)$\displaystyle 4x^3-4xy=0$
    $\displaystyle 4x^3=4xy$
    $\displaystyle x^2=y$

    $\displaystyle 2-2x^2=0$
    $\displaystyle 2=2x^2$
    x=1 & -1

    critical point, = (1,1) & (-1,1) ..is this right???

    realyl hope someone can help, appreciate all help & support.
    Last edited by anderson; Mar 12th 2010 at 06:03 AM.
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  4. #4
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    a: Remember that the two derivatives form a system of equations:
    $\displaystyle x^3-y = 0$
    $\displaystyle y^3-x = 0$

    imply that

    $\displaystyle x^9=x$ or $\displaystyle x(x^8-1) = 0$ (see why?)

    What values for critical points does this give you?

    b: Seems right to me, so far, however, you are being a little hasty. It's quite right that
    $\displaystyle 4x^3-4xy = 0$ implies that
    $\displaystyle x^2 = y$
    but it also implies that
    $\displaystyle x(x^2-y) = 0$. See what you missed?
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  5. #5
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    thank you for replying,

    a)
    $\displaystyle (x^3)^3-x=0$
    $\displaystyle x(x^8-1)$
    x= -1,0 & 1
    critical points=(-1,-1)(0,0) & (1,1)
    b)you are right, but is my final answer for critical point correct?
    critical point=(1,1) & (-1,1) .. i think it has to be correct..not sure,a bit confused though

    is this correct?

    thank you again for all help & support.
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