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Math Help - Extrema if you can

  1. #1
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    Extrema if you can

    Hey i've found this one incredibly tough i don't know if it's just me(probably) but i think its hard and can't seem to get anywhere

    Suppose f(x,y)= A(x2 + Bx + y2 +Cy). What values A, B, C give f(x,y) a local maximum value of 15 at the point (-2, 1)?

    Also find all critical points of
    f(x, y)=ex(1-cos y)

    and classify these critical points.
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  2. #2
    Senior Member Peritus's Avatar
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    <br />
\begin{gathered}<br />
  f(x,y) = A\left( {x^2  + Bx + y^2  + Cy} \right) \hfill \\<br />
   \hfill \\<br />
  \nabla f(x,y) = \left( {2Ax + AB} \right)\hat x + \left( {2By + AC} \right)\hat y = 0 \hfill \\ <br />
\end{gathered} <br />

    <br />
x =  - \frac{B}<br />
{2},y =  - \frac{C}<br />
{2}<br />

    we are told that the local maximum occurs at (-2,1) thus:

    <br />
B = 4,C =  - 2<br />

    next we are told that the value of the local maximum is 15 thus:

    <br />
f(x,y) = \left. {A\left( {x^2  + Bx + y^2  + Cy} \right)} \right|_{(x,y) = ( - 2,1)}  = 15<br />

    so A=-3.

    -------------------------------------------------------------------------------------

    I'm not sure what do you mean by: ex(1-cos y)
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  3. #3
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    f(x,y) = e^x( 1 - cos(y) )
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  4. #4
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    mr fantastic's Avatar
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    Quote Originally Posted by dexza666 View Post
    [snip]
    Also find all critical points of
    f(x, y)=ex(1-cos y)

    and classify these critical points.
    Solve simultaneously:

    \frac{\partial f}{\partial x} = e^x (1 - \cos y) = 0 .... (1)


    \frac{\partial f}{\partial y} = e^x \sin y = 0 .... (2)


    You get (x, \, n \pi) where n is an integer. Now classify in the usual way.
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