His model is incorrect because 20 x 7 is not 14, it is 140. If he had 140 instead of 14, he would have been correct.
I haven't done division like that in a long time, so this is what I think it will look like. I didn't add other work (multiplying 4 with #'s) but you can if you have to
You're looking for the extreme values of subject to
. The Lagrangian is
with critical wherever the partial derivatives vanish:
Substituting the first three solutions into the last equation gives
At these points, we have
so the highest temperature the bee can experience is 28º F at the point (1, 2, -2), and the lowest is -26º F at the point (-1, -2, 2).