# Robin's question at Yahoo! Answers regarding extrema of a function of two variables

#### MarkFL

Staff member
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

Staff member
Hello Robin,

We are given the function:

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

where:

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

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

Let's take a look at a plot of the function on the given domain: Equating the first partials to zero, we obtain:

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

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

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

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

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

Observing that we require:

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

We then have:

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

Thus, we obtain the 5 critical points:

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

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

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

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

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

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

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

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

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

Hence:

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

 Critical point $(a,b)$ $D(a,b)$ $f_{xx}(a,b)$ Conclusion $\left(-\dfrac{\pi}{2},-\dfrac{\pi}{2} \right)$ 1 -1 relative maximum $\left(-\dfrac{\pi}{2},\dfrac{\pi}{2} \right)$ 1 1 relative minimum $(0,0)$ -1 0 saddle point $\left(\dfrac{\pi}{2},-\dfrac{\pi}{2} \right)$ 1 1 relative minimum $\left(\dfrac{\pi}{2},\dfrac{\pi}{2} \right)$ 1 -1 relative maximum