mean value theorem problems

Question

1. does √x(x-1) obey mean value theorem in closed inerval 0,1 .how?

Answer 1

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)+  x^1/2      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