Question

How do you determine all values of c that satisfy the mean value theorem on the interval [0,3] for `f(x) = x^3 + x - 1`?

Answer 1

`sqrt 3`

Explanation:

Mean Value Theorem reads `f(3) = f(0) + 3 f'(c), cin (0, 3)`

`f(x)=x^3+x-1 and f'(x)=3x^2+1`.

`f(3) = 29, f(0)=-1 and f'(c)=3c^2+1`.

So, `29 = -1 + 3(3c^2+1)`

`c^2=3 and c=+-sqrt 3. -sqrt 3` is inadmissible.