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Math Help - Find the max and min

  1. #1
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    Find the max and min

    Find the maximum and minumum values of s^2+t^2 on the curve
    s^2+2t^2-2t=4
    by the method of Lagrange Multipliers.
    If s^2=4-2t^2+2t is substituted into s^2+t^2, we get a function
    h(t)=4+2t-t^2
    which has only a max value on R. Explain how the extreme values you obtained in the first part can be obtained from h(t).

    I only know how to find the max and min from the orgin to to a surface, how do i deal with it when is from s^2+t^2
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  2. #2
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    Quote Originally Posted by wopashui View Post
    Find the maximum and minumum values of s^2+t^2 on the curve
    s^2+2t^2-2t=4
    by the method of Lagrange Multipliers.
    If s^2=4-2t^2+2t is substituted into s^2+t^2, we get a function
    h(t)=4+2t-t^2
    which has only a max value on R. Explain how the extreme values you obtained in the first part can be obtained from h(t).

    I only know how to find the max and min from the orgin to to a surface, how do i deal with it when is from s^2+t^2
    You want to find the extrema of

    f(s,t)=s^2+t^2

    subject to the constraint:

    g(s,t)=s^2+2t^2-2t-4=0.

    So you form the Lagrangian:

    L(s,t,\lambda)=f(s,t)+\lambda g(s,t)

    and the maximum and minimum are amoung the stationary points of L(s,t,\lambda)

    CB
    Last edited by CaptainBlack; October 24th 2010 at 05:47 AM.
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    You want to find the extrema of

    f(s,t)=s^2+t^2

    subject to the constraint:

    g(s,t)=s^2+2t^2-2t-4=0.

    So you form the Lagrangian:

    L(s,t,\lambda)=f(s,t)+\lambdag(s,t)
    You mean L(s,t,\lambda)=f(s,t)+\lambda g(s,t).
    (you don't have a space between "\lambda" and "g".)

    and the maximum and minimum are amoung the stationary points of L(s,t,\lambda)

    CB
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  4. #4
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    so to find this, I need to find the values of x y and lamda, do I let L_x =0, L_y=0, and L_{lamba} =0 to do this
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  5. #5
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    Running through this one's first half, I already found the maximum and minimum values. I'll post them up (but not the steps).

    Max: (s,t)=(\pm 2, 1),s^2+t^2=5
    Min: (s,t)=(0,2),s^2+t^2=4 and (s,t)=(0,-1),s^2+t^2=1

    The second half of this question, however, has me a bit puzzled, as it seems to be a bit of a trick question. The issue I'm having is that h(t) is a parabola that has a maximum at 5 (which matches the first half), though it doesn't have a minimum (or at least no absolute minimum). So how would one find the minimum extrema using h(t)?
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