math

Let f be the function with f(0) = 1/ (pi)^2, f(2) = 1/(pi)^2, and the derivative given by f'(x) = (x+1)cos ((pi)(x)). How many values of x in the open interval (0, 2) satisfy the conclusion of the Mean Value Theorem for the function f on the closed interval [0, 2] ?

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  1. The function is the same at x = 0 and x = 2
    so the mean slope is zero.
    Therefore what it is asking is:
    How many times is f'(x) = (x+1)cos (pi x) = 0
    between pi x = 0 and pi x = 2 pi
    well (x+1) is always a positive number in that interval
    so
    How many times is cos theta = 0 between theta = 0 and theta = 2 pi ????

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    Damon
  2. It is just pretending to be difficult.

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    Damon
  3. So would the answer be pi/2 and 3pi/2?

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