A flat circular plate has the shape of the region x squared plus y squared less than or equals 1.The​ plate, including the boundary where x squared plus y squared equals 1​,is heated so that the temperature at the point (x comma y )is Upper T (x comma y )equalsx squared plus 2 y squared minus three halves x. Find the temperatures at the hottest and coldest points on the plate.

We have the coldest value of temperature . and the hottest value is .

Step-by-step explanation:

We need to take the derivative with respect of x and y, and equal to zero to find the local minimums.

The temperature equation is:

Let's take the partials derivatives.

So, we can find the critical point (x,y) of T(x,y).

The critical point is (3/4,0) so the temperature at this point is:

Now, we need to evaluate the boundary condition.

We can solve this equation for y and evaluate this value in the temperature.

Now, let's find the critical point again, as we did above.

Evaluating T(x,y) at this point, we have:

Now, we can see that at point (3/4,0) we have the coldest value of temperature . On the other hand, at the point we have the hottest value of temperature, it is .

I hope it helps you!