# Thread: Global Extremas

1. ## Global Extremas

Find the global extrema of the function $f(x,y) = sin(xy)$ on the closed region given by $0 \leq\ x \leq \pi$ and $0 \leq\ y \leq 1$. Be sure to clearly indicate the maximum and minimum vlues and where they occur.

Thanks!!

2. I'm still waiting for your partial derivatives and why you think that bore no fruit. Did you investigate those boundaries, yet?

3. Originally Posted by TKHunny
I'm still waiting for your partial derivatives and why you think that bore no fruit. Did you investigate those boundaries, yet?
I tested the boundaries and I think the only critical point I got was $(\pi/2, 1)$.

I found critical points from the partial derivaties and I think the points are $(0,0), (0, 1/2), (\pi, 1/2)$.

I think those are all the critical points.

4. Here's how to classify the critical points. I think you got all of them...how did you get (0,1/2)?

Define $D(a,b) = f_{xx}(a,b) f_{yy} - [f_{xy}(a,b)]^2$

where $(a,b)$ is a critical point.

Here are the rules:

1. If $D > 0$ and $f_{xx}(a,b) > 0$, then $f(a,b)$ is a local minimum

2. If $D > 0$ and $f_{xx}(a,b) < 0$, then $f(a,b)$ is a local maximum

3. If $D < 0$, then $f(a,b)$ is a saddle point (neither a max or min)

those are all the points you got from the boundary?

5. Originally Posted by TKHunny
I'm still waiting for your partial derivatives and why you think that bore no fruit. Did you investigate those boundaries, yet?
I finished the problem and I got...
maximums $(\pi, 1/2)$ and $(\pi/2, 1)$
no minimums

I don't know if this is right.