Finding extrema of f(x,y) = sin(x)sin(y)

[MarkFL]

Here is the question:

How can I find the local maximum and minimum values and saddle points of the function f(x,y) = sin(x)sin(y)?


Where -π < x < π and -π < y < π

I have posed a link there to this thread so the OP can see my work.

 

[MarkFL]

Hello Robin,

We are given the function:

$$f(x,y)=\sin(x)\sin(y)$$

where:

$$-\pi<x<\pi$$

$$-\pi<y<\pi$$

Let's take a look at a plot of the function on the given domain:

View attachment 1676

Equating the first partials to zero, we obtain:

$$f_x(x,y)=\cos(x)\sin(y)=0\implies x=\pm\frac{\pi}{2},\,y=0$$

$$f_y(x,y)=\sin(x)\cos(y)=0\implies x=0,\,y=\pm\frac{\pi}{2}$$

Adding, we find:

$$\sin(x)\cos(y)+\cos(x)\sin(y)=0$$

Applying the angle-sum identity for sine, we find:

$$\sin(x+y)=0$$

Observing that we require:

$$-2\pi<x+y<2\pi$$

We then have:

$$x+y=-\pi,\,0,\,\pi$$

Thus, we obtain the 5 critical points:

$$P_1(x,y)=\left(-\frac{\pi}{2},-\frac{\pi}{2} \right)$$

$$P_2(x,y)=\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$$

$$P_3(x,y)=(0,0)$$

$$P_4(x,y)=\left(\frac{\pi}{2},-\frac{\pi}{2} \right)$$

$$P_5(x,y)=\left(\frac{\pi}{2},\frac{\pi}{2} \right)$$

To categorize these critical points, we may utilize the second partials test for relative extrema:

$$f_{xx}(x,y)=-\sin(x)\sin(y)$$

$$f_{yy}(x,y)=-\sin(x)\sin(y)$$

$$f_{xy}(x,y)=\cos(x)\cos(y)$$

Hence:

$$D(x,y)=\sin^2(x)\sin^2(y)-\cos^2(x)\cos^2(y)$$

[TABLE="class: grid, width: 660"]
[TR]
[TD]Critical point $(a,b)$[/TD]
[TD]$D(a,b)$[/TD]
[TD]$f_{xx}(a,b)$[/TD]
[TD]Conclusion[/TD]
[/TR]
[TR]
[TD]$\left(-\dfrac{\pi}{2},-\dfrac{\pi}{2} \right)$[/TD]
[TD]1[/TD]
[TD]-1[/TD]
[TD]relative maximum[/TD]
[/TR]
[TR]
[TD]$\left(-\dfrac{\pi}{2},\dfrac{\pi}{2} \right)$[/TD]
[TD]1[/TD]
[TD]1[/TD]
[TD]relative minimum[/TD]
[/TR]
[TR]
[TD]$(0,0)$[/TD]
[TD]-1[/TD]
[TD]0[/TD]
[TD]saddle point[/TD]
[/TR]
[TR]
[TD]$\left(\dfrac{\pi}{2},-\dfrac{\pi}{2} \right)$[/TD]
[TD]1[/TD]
[TD]1[/TD]
[TD]relative minimum[/TD]
[/TR]
[TR]
[TD]$\left(\dfrac{\pi}{2},\dfrac{\pi}{2} \right)$[/TD]
[TD]1[/TD]
[TD]-1[/TD]
[TD]relative maximum[/TD]
[/TR]
[/TABLE]