The function in question would be $f: \mathbb{R}^2 \to \mathbb{R}, (x,y) \mapsto x^2y^2 +\sin{y} $. Show that the given set $$ \left \{ \left( 0, \dfrac{\pi}{2} + k\pi \right) : k \in \mathbb{Z} \right \} $$ represents the set of critical points to the question.
Since I'm fairly new to the subject, I feel a bit lost as to how one would go about solving this question. Would it be enough to show that the derivative of $ \dfrac{\pi}{2} + k\pi$ results to 0 or undefined? Or is there something else I'm missing? Thanks in advance!
There are some differences on the definition of critical points depending on the author, but the most common is of a point at which both of the first partial derivatives are zero.
So we wish to solve the system $$\frac{\theta f}{\theta x}=0\\\frac{\theta f}{\theta y}=0$$
In this case we have $$\frac{\theta f}{\theta x}=2xy^2=0\\\frac{\theta f}{\theta y}=2x^2y+\cos y=0$$
The first equation is equivalent to $x=0$ or $y=0$.
For $y=0$ though, the second equation gives $\cos 0=0$ which simply cannot hold since $\cos 0=1$.
Thus, $x=0$, and from the second equation we obtain $\cos y=0 \Rightarrow y=\frac{\pi}{2}+k\pi$.