So I got to the infamous "the proof is left to you as an exercise" of the book when I tried to look up how to get the Lagrange form of the remainder for a Taylor polynomial. Is this right?
Given
$R_{n}(x)=\frac{1}{n!}\int_{0}^{x}f^{n+1}(t)(x-t)^{n}dt$,
find out why
$R_{n}(x)=\frac{1}{(n+1)!}f^{n+1}(c)x^{n+1}$ for some $c\in [0,x]$
According to FTC,
$\int_{0}^{x}f'(t)dt = f(x) – f(0)$
Also, according to the Mean Value Theorem, there exists a $c$ such that
$f'(c)(x-0)=f(x)-f(0)$
so
$\int_{0}^{x}f'(t)dt = f'(c)(x-0)$
finding the derivative of both sides with respect to $x$:
$f'(x) = f'(c)$
so
$f^{n+1}(x) = f^{n+1}(c)$
Going back to the integral form of the remainder:
$R_{n}(x)=\frac{1}{n!}\int_{0}^{x}f^{n+1}(t)(x-t)^{n}dt$,
I replace $f^{n+1}(x)$ with $f^{n+1}(c)$ (This is the step I am most unsure of)
Since $f'(c)$ is a constant, I pull it out of the integral and integrate what's left under the integral, giving me
$R_{n}(x)=\frac{1}{(n+1)!}f^{n+1}(c)x^{n+1}$ for some $c\in [0,x]$
If this is right, then does it mean that $f'(c)$ is the average value of $f'(x)$ from $0$ to $x$?
Sorry if my LaTeX/wording/proof is off. I'd appreciate any corrections/answers to be as simple (notation-wise) as possible please – 1st year undergrad here…
Best Answer
No, it is not right.
$$f'(c) (x-0) = f(x) - f(0)$$ is true, but $c$ depends on $x$.
So it is something like
$$f'(c_x) (x-0) = f(x) - f(0)$$
So derivative of $f'(c) x $ is not really $f'(c)$.
And the step from
$f'(x) = f'(c)$ to $f^{n+1}(x) = f^{n+1}(c)$ is also wrong. If $c$ is constant (according to your proof), then the derivative on the right side becomes $0$.