# Math Help - calculus word problem help

1. ## 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?

2. How would you normally go about minimising/maximising a two-variable function?

3. dont really remember, do I use partial derivatives?

4. 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 $\displaystyle \nabla ^2 T = \left[\begin{matrix}\frac{\partial ^2 T}{\partial x^2} & \frac{\partial ^2 T}{\partial x \partial y} \\ \frac{\partial ^2 T}{\partial y \partial x} & \frac{\partial ^2 T}{\partial y^2} \end{matrix}\right]$.

5. Also, don't forget to check the boundary. The absolute min/max may lie there.

6. Originally Posted by Prove It
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?

7. 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.