Setting both partial derivatives to zero results in a single critical point at
, located within the unit disk.
At this given point, the derivative value of the Hessian matrix is
and the second-order partial derivative with respect to
yields
This suggests that the critical point represents a local minimum, marking it as the coldest area on the plate with a temperature of
.
To find the hottest area on the plate, it must be located along the boundary. Let
and
, so that
Thus, the plate's boundary (the circle
) is treated as a single variable function
examined over
. A single differentiation gives
You will discover that
achieves three extrema on the interval
, with relative maxima occurring at
and
, and a relative minimum at
(and
, if you wish to include that).
Our minimum has already been identified inside the plate - which you can check to have a lower temperature than at the points noted by
- and we identify two maxima at
and
, both showing a maximum temperature of
.
Reverting to Cartesian coordinates, these points match up with
.
