# Hottest and coldest points on a plate given by $T =2x^2−3y^2−2x+10$

I am asking this question because my professor tells me my procedure is not enough to prove my answer. I disagree. That being said, consider the following:

The temperature of a rectangular plate bounded by the lines $$x = ±1,y = ±1$$ is given by $$T =2x^2−3y^2−2x+10$$ Find the hottest and coldest points of the plate.

My solution was very simple: $$t_x=4x-2=0$$$$\Rightarrow x=\frac12$$

$$t_y=-6y=0$$$$\Rightarrow y=0$$ Now, since $$t_{xx}>0, x=\frac12$$ is a minimum, and $$t_{yy}<0, y=0$$ is a maximum. Therefore, $$P_1(\frac12,0)$$ will not be a maximum nor a minimum. Hence, I only had to check the correct combinations. In other words, for example, since $$x=\frac12$$ is a minimum $$y$$ has to be as large as possible if we want to find the minimum. Likewise, at $$y=0$$, $$x$$ has to be as large as possible to maximize the temperature. Intuitively, these points have to occur at the boundary (because of the shape of the graph of $$T$$). Therefore, $$P_2(\frac12,±1)$$ is the coldest point, and $$P_3(-1,0)$$ is the hottest.

My question is, is my procedure enough to prove those points are the hottest and coldest, in other words, is it true in general for quadratic equations?

I know this graph has no saddle points because it lacks $$xy$$ term.

• Are you sure the graph even passes through the rectangular plate? Or am I misunderstanding the question? Commented Dec 6, 2021 at 6:31
• @BrianBlumberg, may I ask you one thing? I am not a college student and I am looking forward to have a career in ML. So once I join college, is this question you posted here one sort of questions that I'll see while learning math as a part of studying for ML?? Commented Dec 6, 2021 at 7:02
• I don't know, but my guess is yes. @Spectre Commented Dec 6, 2021 at 7:07
• @BrianBlumberg thanks then... :) Commented Dec 6, 2021 at 7:07

So you look fine to me. Find all inflection points and then either test them explicitly, or compute second-order derivatives in both $$x$$ and $$y$$.