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Math Help - Is there a way to do this problem without a calculator

  1. #1
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    Is there a way to do this problem without a calculator

    Determine whether the Mean Value Theorem applies to f(x) = 3x – x^2 on the interval [2, 3]. If it can be applied, find all value(s) of c in the interval that satisfy the Mean Value Theorem.

    This is what I did

    \int f(x) dx

    3x^2/2 - x^3/3

    F(3)=3*2^2/2-2^3/3=4.5
    F(2)=3*2^2/2-2^3/3=3+1/3
    4.5-(3+1/3)=7/6

    f(c)=3c-c^2

    I used my graphing calculator and found the intersection between f(c) and 7/6, and the answer is 2.5408.

    Did I do it correctly? If I did, is there a way to do it without a calculator?
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  2. #2
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    Quote Originally Posted by zhangxupage2 View Post
    Determine whether the Mean Value Theorem applies to f(x) = 3x x^2 on the interval [2, 3]. If it can be applied, find all value(s) of c in the interval that satisfy the Mean Value Theorem.

    This is what I did

    \int f(x) dx

    3x^2/2 - x^3/3

    F(3)=3*2^2/2-2^3/3=4.5
    F(2)=3*2^2/2-2^3/3=3+1/3
    4.5-(3+1/3)=7/6

    f(c)=3c-c^2

    I used my graphing calculator and found the intersection between f(c) and 7/6, and the answer is 2.5408.

    Did I do it correctly? If I did, is there a way to do it without a calculator?
    MVT for integrals, yes ...

    \displaystyle 3c - c^2 = \frac{7}{6}

    \displaystyle 0 = c^2 - 3c + \frac{7}{6}

    quadratic formula ...

    \displaystyle x = \frac{3 + \sqrt{9 - \frac{14}{3}}}{2}
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  3. #3
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    Thanks a lot!
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