I have a question which asks me to find the highest and lowest temperatures on a metal plate of radius 5, the temperature at point (x,y) is T(x,y)=4x^2-4xy+y^2
When I take partial derivatives of T with respect to x and y, I get 8x-4y=0 and -4x+2y=0, but this doesn't seem to be much help.
How do I go about solving this problem?
On
With $x > 0, \; z = - y > 0$, this is equivalent to maximising $(2x+z)^2$, where $x^2+z^2= 25$. But by Cauchy-Schwarz inequality and non-negativity of squares, we have
$$125 = (x^2+z^2)(4+1) \ge (2x+z)^2 \ge 0$$ which gives both extrema. The max is when $\frac{x^2}4 = \frac{z^2}1 \implies x = 2\sqrt5, y = -z = -\sqrt5$, and the min is achieved when $x=y=0$.
The critical values lie on the line $y = 2x$. What are the maximum and minimum values of $T(x,2x)$? Do they occur at points $(x,2x)$ inside the disk of radius 5?
The next step is to find the max/min values on the boundary of the disk. You can use a Lagrange multiplier or parameterize the boundary.