Proof of Extended Law of the Mean (Taylor's Formula)

Solution 1:

Let $$F(t)=f(t)-f(x^*)-f'(x^*)(t-x^*)-K(t-x^*)^2$$

where let's set K such that $$0=f(x)-f(x^*)-f'(x^*)(x-x^*)-K(x-x^*)^2$$

So we know that $F(x)=0$ and $F(x^*)=0$

By Rolle's rule, there is a $c1 \in (x, x^*)$ such that $F'(c1)=0$

Now look at the derivative $$F'(t) = f'(t) - f'(x^*)-K \cdot 2(t-x^*)$$

We know $F'(x^*)=0$ and $F'(c1)=0$, so again applying Rolle's theorem, there is a $c2 \in (c1, x^*) \subset (x, x^*)$ such that $F''(c2)=0$

$$F''(t) = f''(t) -2K$$ Therefore $$F''(c2)=0 \rightarrow f''(c2) -2K=0$$

This gives $K=f''(c2)/2$ and you can complete the proof.