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Math Help - Help with the mean value theorem

  1. #1
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    Help with the mean value theorem

    I went in to my professor for "extra help" on this problem and left more confused than before I went in.

    Can someone on here try to explain this to me!

    Question:

    For what values of a, m, and b does the function satisfy the hypotheses of the Mean Value Theorem on the inverval [0, 2]

    Piecewise function:
    f(x)= 3, x=0
    f(x)=-x^2 + 3x + a, 0<x<1
    f(x)=mx + b 1</= x </= 2

    Thank you so much for any help!
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by KarlosK View Post
    I went in to my professor for "extra help" on this problem and left more confused than before I went in.

    Can someone on here try to explain this to me!

    Question:

    For what values of a, m, and b does the function satisfy the hypotheses of the Mean Value Theorem on the inverval [0, 2]

    Piecewise function:
    f(x)= 3, x=0
    f(x)=-x^2 + 3x + a, 0<x<1
    f(x)=mx + b 1</= x </= 2

    Thank you so much for any help!
    In order for the mean value theorem to be applicable, f(x) needs to be continuous on the entire closed interval [0;2], and differentiable in its interior ]0;2[.
    Continuity at x=0 requires a=3, and you can determine the requisite values of b and m from the tangent to the graph of y=-x^2+3x+3 at x_0=1. Because, for f(x) to be differentiable at x_0=1 the graph y=mx+b must be that tangent itself...
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