# Thread: max and min

1. ## max and min

Find the critical point(s) of f(x, y)=sinx+siny+cos(x+y) for 0≤x≤pi/4 and 0≤y≤pi/4. classify each as local max, min or saddle point?

I know the min is 1, but how do you find the max?

2. ## Re: max and min

It is pretty easy to show that there is NO point in the interior of $0\le x \le \pi/4$ and $0\le y\le \pi/4$ where grad f is 0 so any max and min must on the boundary. One boundary is the line x= 0: on that line f(0, y)= sin(y)+ cos(y). Is there any place on that line where f_y(0, y)= 0. Another is the line $x= \pi/4$: on that line $f(\pi/4, y)= \sqrt{2}/2+ sin y+ cos(\pi/4+ y)$. Is there any place on that line where $f_y(\pi/4, y)= 0$? Of course the same thing applies to the lines $y= 0$ and $y= \pi/4$. Any you need to check the values at the corners: $f(0, 0)= 1$, $f(\pi/4, 0)= \sqrt{2}$, $f(0, \pi/4)= \sqrt{2}$, $f(\pi/4, \pi/4)= \sqrt{2}$.