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Math Help - local max. and min.

  1. #1
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    local max. and min.

    Find a formula for a function of the form y = Acos(Bx) + C with a maximum of (0 , 9), a minimum of (5 , 6) and no critical points between these two points.
    A=
    B=
    C=



    need some help with this bad johnny, dont know where to begin...thanks

    mathaction
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  2. #2
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    Quote Originally Posted by mathaction View Post
    Find a formula for a function of the form y = Acos(Bx) + C with a maximum of (0 , 9), a minimum of (5 , 6) and no critical points between these two points.
    ...
    Hello,

    1. calculate the 1rst derivative (use chain rule!):

    f'(x) = -AB\cdot \sin(Bx)

    You know that f'(0) = 0 and f'(5) = 0

    Plug in these values into the equation of the derivative:

    f'(0)=0=-AB\cdot \sin(0) that means you can't determine the values of the parameters.

    f'(5) = 0 = -AB\cdot \sin(5B)~\iff~\sin(5B) = 0~\iff~5B=k\cdot \pi, k\in \mathbb{Z}~\iff~ B=k\cdot \frac \pi5

    Since you know that at x = 5 is the next extremum you know that in this case k = 1 so B=\frac \pi5

    2. Plug in the coordinates of the points you know into the equation of the function:

    f(0)=9=A\cdot \cos\left(\frac \pi5 \cdot 0  \right)+C~\implies~\boxed{9=A+C} because \cos(0) = 1

    f(5) = 6=A\cdot \cos\left(\frac \pi5 \cdot 5  \right)+C~\implies~\boxed{6=-A+C} because \cos(\pi) = -1

    3. Now you have a system of 2 simultaneous equations. Solve for A and C.

    I've got A = 1.5 and C = 7.5

    The equation of your function reads now:

    f(x) = 1.5 \cdot cos\left(\frac \pi5 \cdot x\right)+7.5

    I've attached a diagram of the graph.
    Attached Thumbnails Attached Thumbnails local max. and min.-cos_fktglg.gif  
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  3. #3
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    y = A*cos(Bx) +C

    Max at (0,9) and min at (5,6)

    That means the ampltude A is (1/2)(9-6) = 1.5

    It means also that the perod is 2(5-0) = 10
    So, 10 = 2pi/B
    B = 2pi/10 = pi/5

    It means also that the netral axis s at 6 +1.5 = 7.5
    Meaning, the basic cosine curve is shifted vertically 7.5 units up.
    Meaning, C = 7.5

    Therefore, y = (1.5)cos[(pi/5)x] +7.5 ---------------answer.

    Check for max y. It s at (0,9):
    y = (1.5)cos[(pi/5)*0] +7.5
    y = (1.5)cos(0) +7.5
    y = 1.5(1) +7.5
    y = 9
    So, (0,9) ----checks

    For min, at (5,6):
    y = (1.5)cos[(pi/5)(5)] +7.5
    y = (1.5)cos(pi) +7.5
    y = 1.5(-1) +7.5
    y = 6
    So, (5,6) ----checks again.
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