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Math Help - calculus word problem help

  1. #1
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    calculus word problem help

    The temperature in the steam room of the Star Ship Enterprise is given by the following function:

     T(x,y) = 100 - 4x^{2} -y^{2} + x^{2}y^{2}-1

     -1 \leq x \leq 1

     -2 \leq y \leq 3

    Captain Kirk likes the hottest temperature while Mr Spock prefers the coolest area of the steam room.
    Where should each of them sit? Why?
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  2. #2
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    How would you normally go about minimising/maximising a two-variable function?
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    dont really remember, do I use partial derivatives?
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    Yes, you first need to find where \displaystyle \nabla T = \left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y}\right) = \mathbf{0}, and then evaluate the Hessian at those points \displaystyle \nabla ^2 T = \left[\begin{matrix}\frac{\partial ^2 T}{\partial x^2} & \frac{\partial ^2 T}{\partial x \partial y} \\ \frac{\partial ^2 T}{\partial y \partial x} & \frac{\partial ^2 T}{\partial y^2} \end{matrix}\right].
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  5. #5
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    Also, don't forget to check the boundary. The absolute min/max may lie there.
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  6. #6
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    Quote Originally Posted by Prove It View Post
    Yes, you first need to find where \displaystyle \nabla T = \left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y}\right) = \mathbf{0}, and then evaluate the Hessian at those points \displaystyle \nabla ^2 T = \left[\begin{matrix}\frac{\partial ^2 T}{\partial x^2} & \frac{\partial ^2 T}{\partial x \partial y} \\ \frac{\partial ^2 T}{\partial y \partial x} & \frac{\partial ^2 T}{\partial y^2} \end{matrix}\right].

    I am not quite sure what to do yet, do I find the partial derivative with respect to x and y, than set it equal to zero?
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    First, you need to find all the first and second partial derivatives.

    Set the first partial derivatives equal to \displaystyle 0 and solve for \displaystyle x, y. Then substitute this value into the second partial derivatives (in the Hessian Matrix). If your Hessian is postive definite, you have a minimum. If it's negative definite, you have a maximum.

    And like Danny said, you also need to check the boundaries.
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