Finding maximal and minimal temperature on barbed wire

Question

A barbed wire has circle form $x^2+y^2-2y=0$. The temperature on a point on the wire is given by the function $t(x,y)=2x^2+3y$.

Find minimal and maximal temperature points on the wire.

How does one solve this?

Thanks!

Answer 1

The answer are Lagrange multipliers or in this case just a clever look:

In the area where you are looking for both extremes, it holds $x^2 = 2y-y^2$. That means that $x$ depends on $y$ in your area, therefore your temperature $t$ can be viewed as a function of one variable: $$t(x,y) = t(x(y),y) = 2(2y-y^2)+3y = -2y^2+7y = -2(y-\frac{7}{4})^2-\frac{49}{8}\text{.}$$ You should know how to proceed.

EDIT:

From now on I assume that $t$ is a function of one variable. I should point out that it could happen that $y_0$ which satisfies the equality $t'(y_0) = 0$ does not lie in your area. This would be true if $$2y_0 - y_0^2<0\text{.}$$ In that case you cannot find corresponding $x_0$, because it would hold that $x_0^2 <0$. However, this is not the case, since $2y_0 - y_0^2=\frac{7}{4}$, therefore extreme of $t$ may be at points $(\pm \frac{\sqrt{7}}{4},\frac{7}{4})$.

You said you had problems finding minimums, so I suspect that you believe that extreme of a function can be achieved only in it's stationary points, which is wrong. If a differentiable function $$g:[a,b]\to\mathbb{R}$$ is given, it can achieve it's extrem in any of it's stationary points or in the borders of interval.

So what is the algorithm for finding global extremes?

1. Find interval $[a,b]$ where $t$ is defined.
2. Find all candidates (stationary points + border points)
3. Compute values of function in the points from 2.

Now, you should really know how to proceed.

Question: what would happend if $g$ is defined on $[a,b]\cup [c,d]$, where $b<c$? Can you find extreme(s) of such function?