# Mean Value Theorem/Examples/x^3/Formulation 2

## Example of Use of Mean Value Theorem

Let $f$ be the real function defined as:

$\map f x = x^3$

Let:

$c = 2$, $h = -1$

Then when $\theta = 2 - \sqrt {\dfrac 7 3}$:

$\map {f'} {c + \theta h} = \dfrac {\map f {c + h} - \map f c} h$

## Proof

We have:

 $\ds \map f {c + h}$ $=$ $\ds \map f 1$ $\ds$ $=$ $\ds 1$ $\ds \map f c$ $=$ $\ds 8$ $\ds \leadsto \ \$ $\ds \dfrac {\map f {c + h} - \map f c} h$ $=$ $\ds \dfrac {1 - 8} {-1}$ $\ds \leadsto \ \$ $\ds \map {f'} {c + \theta h}$ $=$ $\ds 7$ Mean Value Theorem $\ds$ $=$ $\ds 3 \paren {c + \theta h}^2$ Power Rule for Derivatives $\ds \leadsto \ \$ $\ds 2 - \theta$ $=$ $\ds \sqrt {\dfrac 7 3}$ $\ds \leadsto \ \$ $\ds \theta$ $=$ $\ds 2 - \sqrt {\dfrac 7 3}$

and we see that:

$1 < 2 - \sqrt {\dfrac 7 3} < 2$

$\blacksquare$