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Inflection points, Intervals and minimum value of a Function

  1. Nov 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider the function below. (Round the answers to two decimal places. If you need to use - or , enter -INFINITY or INFINITY.)
    f(θ) = 2cos(θ) + (cos (θ))^2
    0 ≤ θ ≤ 2π
    (a) Find the interval of increase.
    ( , )

    Find the interval of decrease.
    ( , )

    (b) Find the local minimum value.


    (c) Find the inflection points.
    ( , ) (smaller x value)
    ( , ) (larger x value)

    Find the interval where the function is concave up.
    ( , )

    Find the intervals where the function is concave down. (Enter the interval that contains smaller numbers first.)
    ( , ) ( , )

    2. Relevant equations
    f(θ) = 2cos(θ) + (cos (θ))2
    0 ≤ θ ≤ 2π


    3. The attempt at a solution
    I tried using sign analysis with the second derivative (I definitely know that you have to use the second derivative) and all of the values for it. Thus:
    _____________-2sinx 1 + cosx f'(x) f(x)
    0 = 0
    0 < x < pi/2
    x = pi/2
    pi/2 < x < pi
    x = pi
    pi < x < 3pi/2
    x = 3pi/2
    3pi/2 < x < 2pi
    x = 2 pi

    But I honestly don't know where to go from there


    1. The problem statement, all variables and given/known data
    Consider the function below. (Give your answers correct to two decimal places. If you need to use - or , enter -INFINITY or INFINITY.)
    f(x) = e^[-1/(x + 2)]

    (c) Find the inflection point.
    ( , )

    Find the intervals where the function is concave up.
    ( , ) ( , )

    2. Relevant equations
    f(x) = e^[-1/(x + 2)]

    3. The attempt at a solution
    I tried something similar to the above question
     
  2. jcsd
  3. Nov 30, 2009 #2

    Mark44

    Staff: Mentor

    You have f(t) = 2cos(t) + cos^2(t) -- (I'm using t instead of theta)

    I don't see where you have found f'(t) or f''(t). You need both of those derivatives in order to talk about where f is increasing/decreasing and where the inflection points are.
     
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