Finding stationary points of a variable of two functions
The given function is $f(x, y) = x \sin y$. I've determined the gradient of $f$ to be $ \nabla f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle \to \langle \sin y, x \cos y \rangle $.
To be able to find stationary points, I would have to set $ \nabla f(x, y) = \vec{0} = \langle 0, 0 \rangle $.
$$ \nabla f(x, y) = \vec{0} $$ $$ \langle \sin y, x \cos y \rangle = \langle 0, 0 \rangle $$ $$ \sin y = 0 \;{\mathrm {and}}\; x \cos y = 0 $$
This is where I would be stuck. Which equation should I use to find stationary points?
$\endgroup$1 Answer
$\begingroup$We want to simultaneously solve
$$\sin y = 0 \\ x \cos y = 0$$
From the first equation
$$y = \pi~ n, n \in \mathbb{Z}$$
Using that result in the second equation
$$x \cos(\pi~ n) = 0 \implies x = 0$$
So, the stationary points are
$$(x, y) = (0, \pi ~n), n \in \mathbb{Z}$$
Here is a graphical representation using a contour plot
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
Here is a 3D plot
Notice the waves due to the periodic function and the repeating pattern. Look along $x=0$ and note the $y=\pi~n$ locations.
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