Question

Does the function satisfy the hypotheses of the mean value theorem on the given interval? f(x) = x3 + x − 9, [0, 2]

Answer 1

$\frac{f(2)-f(1)}{2-1}= \frac{2^3+2-9-(1^3+1-9)}{1}=8+2-9-1-1+9=8 \\ derivative of f(x) =3x^2+1,3x^2+1=8,3x^2=7,x= \sqrt \frac{7}{3}$
which supports the mean value theorem, you want even better news
if the function is a polynomial it will always follow the mean value theorem