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
.
At this given point, the derivative value of the Hessian matrix is
and the second-order partial derivative with respect to
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
Thus, the plate's boundary (the circle
You will discover 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
Reverting to Cartesian coordinates, these points match up with
