1. ## expansion of function..?

I've found the following statement in my textbook:

If $\displaystyle f: I \subset \mathbb{R} \rightarrow \mathbb{R}$ is a function of class $\displaystyle C^2$, than there exists $\displaystyle \varepsilon \in <0, 1>$ such that $\displaystyle f(x+h)=f(x) + h f'(x) + \frac{h^2}{2} f'' (x+\varepsilon h)$

This is not a consequence of anything we've done in the lesson, so I was hoping someone could tell me is what this expansion is.

Thank you!

(It reminds me of Taylor series, but it's not, I think..)

2. Originally Posted by marianne
I've found the following statement in my textbook:

If $\displaystyle f: I \subset \mathbb{R} \rightarrow \mathbb{R}$ is a function of class $\displaystyle C^2$, than there exists $\displaystyle \varepsilon \in <0, 1>$ such that $\displaystyle f(x+h)=f(x) + h f'(x) + \frac{h^2}{2} f'' (x+\varepsilon h)$

This is not a consequence of anything we've done in the lesson, so I was hoping someone could tell me is what this expansion is.

Thank you!

(It reminds me of Taylor series, but it's not, I think..)
it is Taylor series indeed: $\displaystyle f(b)=f(a)+(b-a)f'(a) + \frac{(b-a)^2}{2}f''(c),$ for some $\displaystyle c$ between $\displaystyle a$ and $\displaystyle b.$ thus $\displaystyle c= (1-\epsilon)a+\epsilon b,$ for some $\displaystyle 0<\epsilon<1.$ now put $\displaystyle b=x+h$ and $\displaystyle a=x.$

3. Thank you!!!