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Math Help - Find the extrema of f(x) = 2 sinx + cos 2x when 0 < x < 2pi

  1. #1
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    Find the extrema of f(x) = 2 sinx + cos 2x when 0 < x < 2pi

    I am having trouble determining the extrema of the above function. I think the derivative is 2 cos x - 2 sin 2x At 2 cos x= 0 ; cos is pi/2 and/or 3 pi/2. At - 2 sin 2x = 0; - 2 sin 2x = pi and/or 2 pi. 2 cos x = pi, ---> cos x = pi/2 and/or 3 pi/4. - 2 sin 2x = -pi/2 ---> - pi/4 or - pi/2. If what I did so far is correct, I do not understand how to use the information to determine the extrema of x a 0 less than or equal than x less than or equal than 2 pi.
    Last edited by bosmith; April 20th 2009 at 08:08 AM. Reason: I left out the last sentence.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by bosmith View Post
    I am having trouble determining the extrema of the above function. I think the derivative is 2 cos x - 2 sin 2x At 2 cos x= 0 ; cos is pi/2 and/or 3 pi/2. At - 2 sin 2x = 0; - 2 sin 2x = pi and/or 2 pi. 2 cos x = pi, ---> cos x = pi/2 and/or 3 pi/4. - 2 sin 2x = -pi/2 ---> - pi/4 or - pi/2
    I would factor the derivative:
     2 * cos x - 2 * 2 *sin x cos x = 0

    into:
     2 * cos x ( 1 - 2* sin x) = 0

    and solve for the critical numbers. make sure you justify what type of extrema you find, using one of the derivative tests.

    Good luck!!
    Last edited by apcalculus; April 20th 2009 at 08:10 AM. Reason: add math tags
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  3. #3
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    Thank you for the derivative 2 cos x ( 1 - 2 sin x)

    I came up with critical numbers of pi/4, 7 pi/6 and 11 pi/6. How do these numbers relate to 0 less than or equal to x less than or equal to 2 pi? The smallest is pi/4 or 45 degrees, the largest is 11 pi/6 or 330 degrees. I appreciate your help. I truly need to understand how to solve this problem
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  4. #4
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    This message is to apcalculus. I amj a little nervous and may be replying wrongly

    I came up with critical numbers of pi/4, 7 pi/6 and 11 pi/6. How do these numbers relate to 0 less than or equal to x less than or equal to 2 pi? The smallest is pi/4 or 45 degrees, the largest is 11 pi/6 or 330 degrees. I appreciate your help. I truly need to understand how to solve this problem
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  5. #5
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by bosmith View Post
    I came up with critical numbers of pi/4, 7 pi/6 and 11 pi/6. How do these numbers relate to 0 less than or equal to x less than or equal to 2 pi? The smallest is pi/4 or 45 degrees, the largest is 11 pi/6 or 330 degrees. I appreciate your help. I truly need to understand how to solve this problem

    cos x = 0

    OR

    1 - 2 sin x = 0

    In the interval (0, 2Pi), the first condition gives x = Pi/2 or x = 3Pi/2

    The second condition says sin x = 1/2 which happens at 30 degrees and 150 degrees. These two angles are: Pi/6 and 5Pi/6, respectively.

    So, the critical numbers in your open interval (0, 2Pi) are, in increasing order:

    Pi/6, Pi/2, 5Pi/6, and 3Pi/2

    at Pi/6: the f' changes sign from + to - (graph the first derivative to see this!) so it is a max

    at Pi/2: the f' changes sign from - to + so it is a min

    at 5Pi/6: the f' changes sign from + to - so it is a max

    and finally:

    at 3Pi/2: the f' changes sign from - to + so it is a min.


    I hope this helps!!
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  6. #6
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    Finding critical numbers for a trigonometric derivative

    apcalculus. Yes, you helped me very much. Thank you.
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