Question

How to find all the critical points of f(x,y) = sinx cosy?

Answer 1

critical points are the points when the partial derivatives of the function are "0" simultaneously

Answer 2

f(x,y) = sinx cosy
\[\nabla f =(\cos x \cos y, -\sin x\sin y) \]
So, we get
cosx cosy = 0
-sinx siny = 0

But I am not sure how to solve this system of equations.

Answer 3

\[
\begin{align}
f_x&=\cos x\,\cos y=0\qquad{\rm and}\\
f_y&=-\sin x\,\sin y
\end{align}
\]
both the sin and cos are periodic functions
notice that when \(x=2\pi n\), \(\sin x=0\)

Answer 4

similarly, when \(y=2\pi n-{\pi\over2}\), \(\cos y=0\)

Answer 5

this is one pair of "x" and "y" to give the critical points.
there should be in all \(2^3\) such pairs.