Aflat circular plate has the shape of the region x squared plus y squared less than or equals 1x2+y2≤1. the plate, including the boundary where x squared plus y squared equals 1x2+y2=1, is heated so that the temperature at the point left parenthesis x comma y right parenthesis(x, y) is upper t left parenthesis x comma y right parenthesist(x, y)equals=x squared plus 3 y squared plus one third xx2+3y2+13x. find the temperatures at the hottest and coldest points on the plate.

You're looking for the extreme values of subject to the constraint .

The target function has partial derivatives (set equal to 0)

so there is only one critical point at . But this point does not fall in the region . There are no extreme values in the region of interest, so we check the boundary.

Parameterize the boundary of by

with . Then can be considered a function of alone:

has critical points where :

but for all , so this case yields nothing important.

At these critical points, we have temperatures of

so the plate is hottest at (1, 0) with a temperature of 14 (degrees?) and coldest at (-1, 0) with a temp of -12.