So, I was going through A First Course in Analysis by John B. Conway, Springer, first edition (2018) and came across this question in a section (6.6 to be precise) talking about the critical points of functions $f: \mathbb{R}^n \rightarrow \mathbb{R}$;
Let $\Omega = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 < 1\}$ and define $f$ on $\Omega$ by $f(x,y,z) = \sin(\pi x)^2 + \cos(\pi y)^2 - z^2$. Find all the critical points of $f$ in $\Omega$ and give the nature of each whenever you can.
According to some of my friends and my professor, there are supposedly 25 critical points. However, in all my efforts I have only been able to uncover 9 such points. Below is a summation of results of my efforts, and what I know;
Let $\Omega$ and $f$ be defined as above. Then to find our critical points, we must find all $a \in \mathbb{R}^3$ such that $\nabla f(a) = 0$, where $\nabla f(a) = \left(\frac{\partial f}{\partial x}(a), \frac{\partial f}{\partial y}(a), \frac{\partial f}{\partial z}(a) \right)$.
So, we can find that $$\partial_x f = 2\pi \sin(\pi x)\cos(\pi x) $$ $$ \partial_y f = -2\pi \sin(\pi y)\cos(\pi y)$$ $$ \partial_z f = -2z $$ and setting each $\partial f = 0$, we can find that $$\partial_x f = 0\text{ when } x = 0, \frac{1}{2}, -\frac{1}{2}$$ $$\partial_y f = 0\text{ when } y = 0, \frac{1}{2}, -\frac{1}{2}$$ $$\partial_z f = 0\text{ when } z = 0$$
So, our possible critical points are \begin{array}[ccc] ((0,0,0) & (0,\frac{1}{2},0) & (0,-\frac{1}{2},0) \\ (\frac{1}{2},0,0) & (\frac{1}{2},\frac{1}{2},0) & (\frac{1}{2},-\frac{1}{2},0)\\ (-\frac{1}{2},0,0) & (-\frac{1}{2},\frac{1}{2},0) & (-\frac{1}{2},-\frac{1}{2},0)\end{array}
So, by the second derivative test, we can determine the nature of the critical points listed above; $$ \text{det}\left( D^2f(0,0,0) \right) = 8\pi^4 \implies (0,0,0) \text{ is a local min.} $$ $$ \text{det}\left( D^2f(0,\frac{1}{2},0) \right) = -8\pi^4 \implies (0,\frac{1}{2},0) \text{ is a saddle point} $$ $$ \text{det}\left( D^2f(0,-\frac{1}{2},0) \right) = -8\pi^4 \implies (0,-\frac{1}{2},0) \text{ is a saddle point} $$ $$ \text{det}\left( D^2f(\frac{1}{2},0,0) \right) = -8\pi^4 \implies (\frac{1}{2},0,0) \text{ is a saddle point} $$ $$ \text{det}\left( D^2f(\frac{1}{2},\frac{1}{2},0) \right) = 8\pi^4 \implies (\frac{1}{2},\frac{1}{2},0) \text{ is a local min.} $$ $$ \text{det}\left( D^2f(\frac{1}{2},-\frac{1}{2},0) \right) = 8\pi^4 \implies (\frac{1}{2},-\frac{1}{2},0) \text{ is a local min.} $$ $$ \text{det}\left( D^2f(-\frac{1}{2},0,0) \right) = -8\pi^4 \implies (-\frac{1}{2},0,0) \text{ is a saddle point} $$ $$ \text{det}\left( D^2f(-\frac{1}{2},\frac{1}{2},0) \right) = -8\pi^4 \implies (-\frac{1}{2},\frac{1}{2},0) \text{ is a saddle point} $$ $$ \text{det}\left( D^2f(-\frac{1}{2},-\frac{1}{2},0) \right) = 8\pi^4 \implies (-\frac{1}{2},-\frac{1}{2},0) \text{ is a local max.} $$ given that $\partial_{xx}f(0,0,0) = \partial_{xx}f(0,\frac{1}{2},0) = \partial_{xx}f(0,-\frac{1}{2},0) = 2\pi^2 > 0$ and $\partial_{xx}f(\frac{1}{2},0,0) = \partial_{xx}f(\frac{1}{2},\frac{1}{2},0) = \partial_{xx}f(\frac{1}{2},-\frac{1}{2},0) = \partial_{xx}f(-\frac{1}{2},0,0) = \partial_{xx}f(-\frac{1}{2},\frac{1}{2},0) = \partial_{xx}f(-\frac{1}{2},-\frac{1}{2},0) = -2\pi^2 < 0$.
And one last thing to clarify a little more of what I think I understand about this: So to the best of my knowledge, the second derivative test involves using the Hesian of $f(a)$, which means just finding the determinant of the matrix created from the second derivatives of $f$. Although, I haven't heard the word Hesian anywhere other than online once I started looking for help. Also, one of my friends said that due to the fact that $z$ must always be zero to have a critical point there, something special happens, but I don't understand EXACTLY what he meant by that. It makes sense but still doon't know how to put it down on paper.
So finally, here are my questions:
1.) How would I go about finding the other 16 critical points (if there are truly that many)? Does it involve looking at the eigenvalues of each $D^2f(a)$ where $a$ is a potential critical point?
2.) In the future, how would I know that there even are that many more critical points to the function? Does it have to do with the cyclic nature of sine and cosine?
I know this was a long post but I've been working all week on this and just want to understand what I need to do! Thank you in advance!