# calculus word problem help

• Jan 18th 2011, 03:10 AM
Tweety
calculus word problem help
The temperature in the steam room of the Star Ship Enterprise is given by the following function:

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

$\displaystyle -1 \leq x \leq 1$

$\displaystyle -2 \leq y \leq 3$

Captain Kirk likes the hottest temperature while Mr Spock prefers the coolest area of the steam room.
Where should each of them sit? Why?
• Jan 18th 2011, 03:12 AM
Prove It
How would you normally go about minimising/maximising a two-variable function?
• Jan 18th 2011, 03:25 AM
Tweety
dont really remember, do I use partial derivatives?
• Jan 18th 2011, 03:42 AM
Prove It
Yes, you first need to find where $\displaystyle \displaystyle \nabla T = \left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y}\right) = \mathbf{0}$, and then evaluate the Hessian at those points $\displaystyle \displaystyle \nabla ^2 T = \left[\begin{matrix}\frac{\partial ^2 T}{\partial x^2} & \frac{\partial ^2 T}{\partial x \partial y} \\ \frac{\partial ^2 T}{\partial y \partial x} & \frac{\partial ^2 T}{\partial y^2} \end{matrix}\right]$.
• Jan 18th 2011, 04:36 AM
Jester
Also, don't forget to check the boundary. The absolute min/max may lie there.
• Jan 18th 2011, 06:51 AM
Tweety
Quote:

Originally Posted by Prove It
Yes, you first need to find where $\displaystyle \displaystyle \nabla T = \left(\frac{\partial T}{\partial x},\frac{\partial T}{\partial y}\right) = \mathbf{0}$, and then evaluate the Hessian at those points $\displaystyle \displaystyle \nabla ^2 T = \left[\begin{matrix}\frac{\partial ^2 T}{\partial x^2} & \frac{\partial ^2 T}{\partial x \partial y} \\ \frac{\partial ^2 T}{\partial y \partial x} & \frac{\partial ^2 T}{\partial y^2} \end{matrix}\right]$.

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?
• Jan 18th 2011, 07:03 AM
Prove It
First, you need to find all the first and second partial derivatives.

Set the first partial derivatives equal to $\displaystyle \displaystyle 0$ and solve for $\displaystyle \displaystyle x, y$. 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.