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Math Help - max and min

  1. #1
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    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?
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  2. #2
    MHF Contributor

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    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}.
    Last edited by HallsofIvy; February 21st 2013 at 01:30 PM.
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