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Math Help - please help me with the Mean Value Theorem...

  1. #1
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    please help me with the Mean Value Theorem...

    Hello,

    I'm having trouble with this maths question and was wondering if someone could help me?? I've been asked to use the Mean Value Theorem to show that:

    sqrt (x*y) < 1/2 * (x + y) if 0 < x < y

    And we were hinted to use the function f(x) = sqrt x

    I'm really confused at where to start... please help me...
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  2. #2
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    Given the interval [x,y] for the function f(z) = \sqrt{z} (so I don't confuse the variable of the function with the endpoint of my interval), there exists some c such that:
    f'(c) = \frac{f(y) - f(x)}{y-x} = \frac{\sqrt{y} - \sqrt{x}}{y - x}

    Since 0 < x < y, we know that f'(c) > 0.

    Now, the trick is to multiply top and bottom by "1", that is: \sqrt{y} - \sqrt{x}. So we get:

     f'(c) = \frac{\sqrt{y} - \sqrt{x}}{y-x} \cdot \left(\frac{\sqrt{y} - \sqrt{x}}{\sqrt{y} - \sqrt{x}}\right) > 0

    f'(c) = \frac{y - 2\sqrt{xy} + x}{\mbox{(Some denominator)}} > 0

    Multiply both sides by the denominator and you'll (hopefully) reach your inequality.
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  3. #3
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    Thankyou so much!! You really helped me!!
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