Let be a differentiable function with continuous derivative such that . If

(1)

then prove that there exists a such that

**Solution**

Consider the function

which is clearly continuous on , since:

Furthermore, is differentiable in . The derivative is given by

Since it follows from Rolle’s theorem that there exists a such that . The result now follows.

As a side note: If we consider the function

which is continuous in and differentiable in then we get that there exists a such that

Coincedence?