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Math Help - max. and min.

  1. #1
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    max. and min.

    Let f(x) = |x^2 - 1 | x is an element of [0,2]
    a) find were f is strictly increasing and where it is strictly decreasing
    b) find the maximum and minimum of f on [0,2]
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  2. #2
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    Quote Originally Posted by slowcurv99 View Post
    Let f(x) = |x^2 - 1 | x is an element of [0,2]
    b) find the maximum and minimum of f on [0,2]
    The function f(x)=|x^2-1| on [0,2] is continous and has a maximum and minimum.

    Find the critical points, i.e. where the derivative does not exists or is zero. The place where the derivative does not exist can only occur at x^2-1=0 that is x=1 on [0,2]. I leave it to thee to show the function is not differenciable there.

    Next find the derivative of f(x). To do that use the following theorem.

    Theorem: Let f(x)=|x| then if x!=0 the function is differenciable and the derivative is sgn(x). Here.

    Thus, the derivative of f(x)=|x^2-1| on (0,2) and x!=1 is:
    2x*sgn(x^2-1)
    We need that,
    2x*sgn(x^2-1)=0
    Since sgn(x^2-1)!=0 for x!=0 we must have,
    2x=0.
    Which has no solutions on (0,2).
    Thus, the only critical point is the non-differenciable point we found before x=1.

    Compare values with x=0 and x=2. To find maximum and minimum.
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