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Increase and Decrease of a function question

  1. Jan 12, 2012 #1
    1. The problem statement, all variables and given/known data
    f(x)=x — cosx, 0 ≤ x ≤ 2∏


    2. Relevant equations



    3. The attempt at a solution

    I took the derivative of the function: f'(x)=1+ sinx, 0 ≤ x ≤ 2∏
    I don't know how to determine whether it is increasing or decreasing between the interval given. What should I do?
     
  2. jcsd
  3. Jan 12, 2012 #2

    SammyS

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    What do you know about the range of sin(x) ?
     
  4. Jan 12, 2012 #3
    The Range of sin(x) is -1≤ x ≤ 1?

    I have an additional question:

    I saw this in my textbook and don't understand why this is true.

    f (x) = -2sin2x — 2sinx = -2sinx(2cosx+1) <— how is this true?
    Thanks
     
  5. Jan 12, 2012 #4

    SammyS

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    So... Have you solve the original problem?

    As for

    f (x) = -2sin2x — 2sinx
    = -2sinx(2cosx+1)​
    The double angle identity for sine is: sin(2x) =2sin(x)cos(x)

    Therefore: -2(2sin(x)cos(x)+sin(x)) =  ? 
     
  6. Jan 12, 2012 #5
    No. I drew the graph and it doesn't always increase but the answer says it increases from [0,2∏] How does knowing the range of sine solve help?

    Actually, I've never learned "The double angle identity". I guess I have to know this then.
     
  7. Jan 12, 2012 #6

    SammyS

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    Well, you got the derivative: f'(x)=1+ sin(x).

    -1 ≤ sin(x) ≤ 1

    Therefore:

    0 ≤ 1+ sin(x) ≤ 2

    So f' is positive except where sin(x) = -1, correct ?

    Take it from there.

    (It's a little tricky to figure out the behavior where f' is zero.)
     
  8. Jan 12, 2012 #7
    Thank you SammyS.
     
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