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

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 = 1/3According 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