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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?

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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

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$enter image description here

Here is a 3D plot

enter image description here

Notice the waves due to the periodic function and the repeating pattern. Look along $x=0$ and note the $y=\pi~n$ locations.

  • 0
    Going off that stationary point, is it correct to say that there will be many local maxima and local minima? My reasoning is that I'm assuming the graph of $z = x \sin y $ has waves in it. – 2017-02-15
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    I added a 3D Plot so you can see the waves and the repeating pattern. Look along $x = 0$ and note the $y = \pi~ n$ locations. – 2017-02-15