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    Functions and calculus

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    (See attached file for full problem description)

    1) Consider the following function:
    a) f (x) = 9x2 - x3

    b) f (x) = x + 1
    x - 2

    c) f (x) = x2/3 (x - 5)

    for each of the above functions complete the following table. Show the work to justify your answers below the table.

    f(x) is increasing on
    f(x) is decreasing on
    Rmax of f(x) is at
    Rmin of f(x) is at
    x-intercept of f(x) is at
    y-intercept of f(x) is at
    f(x) is C. U. on
    f(x) is C. D. on
    P. I. of f(x) is at

    2) Let f(x) = x + sin 2x on [0, 2 pie] find two numbers (c ) that satisfy the conclusion of the Mean Value Theorem. ( note there are four such numbers)

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    Solution Preview

    Please refer to the attached file. Cheers.

    1) Consider the following function:
    a) f (x) = 9x2 - x3

    b) f (x) = x + 1
    x - 2

    c) f (x) = x2/3 (x - 5)


    f(x) is increasing when f'(x) > 0.

    f(x) is decreasing when f'(x) < 0.

    The critical points are when f'(x) = 0

    Rmax is at (6,108)
    Rmin is at (0,0)

    x-intercept occurs when y = 0

    x-intercepts are at x =0 and x = 9

    y-intercept occurs when x = 0

    y-intercept is at (0,0)

    f(x) is concave up when f"(x) > 0

    f(x) is concave down when f"(x) 0

    Point of ...

    Solution Summary

    This shows how to determine max, min, intercepts, concavity, and increasing/decreasing for functions.