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Math Help - Extremum of f

  1. #1
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    Extremum of f

    If f(x,y) = sin x+ siny + sin (x+y), 0\le x\le2\pi, 0\le y \le 2\pi, find the extremum of f.

    I find f_x and f_y and equate them to zero. Then from these two equations, I get cos x= cos y.

    Can anyone help me to proceed from here to get the extremum?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by problem View Post
    If f(x,y) = sin x+ siny + sin (x+y), 0\le x\le2\pi, 0\le y \le 2\pi, find the extremum of f.

    I find f_x and f_y and equate them to zero. Then from these two equations, I get cos x= cos y.

    Can anyone help me to proceed from here to get the extremum?
    With both $$x and $$y in [0,2\pi], \cos(x)=\cos(y) implies that $$x=$$y, which will imply something else ...

    CB
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  3. #3
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    Not quite. x, y in [0, 2\pi] and cos(x)= cos(y) implies either x= y or x= 2\pi- y. Put those into the two equations cos(x)+ cos(x+y)= 0 and cos(y)+ cos(x+y)= 0.

    (Although I don't believe that second case gives a new solution.)
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  4. #4
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    By getting x=y or x=2\pi-y, we still can not imply anything for a critical value as we have x and y in between 0 and 2\pi.
    Is it that we still have to create another equation to get the critical value?

    By the way, the answer given is x=y=\frac{5\pi}{3} is the minimum and x=y=\frac{\pi}{3} is the maximum value.
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  5. #5
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    Quote Originally Posted by problem View Post
    By getting x=y or x=2\pi-y, we still can not imply anything for a critical value as we have x and y in between 0 and 2\pi.
    Is it that we still have to create another equation to get the critical value?
    If x= y, then sin(x)+ sin(y)+ sin(x+ y)= sin(x)+ sin(x)+ sin(x+x)= 2sin(x)+ sin(2x).

    By the way, the answer given is x=y=\frac{5\pi}{3} is the minimum and x=y=\frac{\pi}{3} is the maximum value.
    I suggest you check that again. The values of x and y are NOT values of the function and cannot be maximum and minimum values for the function.
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by problem View Post
    By getting x=y or x=2\pi-y, we still can not imply anything for a critical value as we have x and y in between 0 and 2\pi.
    Is it that we still have to create another equation to get the critical value?

    By the way, the answer given is x=y=\frac{5\pi}{3} is the minimum and x=y=\frac{\pi}{3} is the maximum value.
    What is f_x(x,y) (and/or f_y(x,y)), now that you know that x=y or x=2\pi-y what do the equations f_x(x,y)=0 and f_y(x,y)=0 become in each of these cases?

    CB
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