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mean value theorem problems

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If the factor (x-1) is under the square root , then the mean value theorem does not hold because the function is not defined on the interior points (because it would involve taking the square root of a negative number).
If the factor (x-1) is outside the square root, then the MVT does apply.  By the product rule the derivative is:
(1/2)x-1/2 (x-1)+  x1/2      This equals zero when x = 1/3
According to the MVT   the derivative must equal the endpoint to endpoint slope (which is zero) at some point in the interior.    In this case that interior point is x = 1/3