After looking over a few statements and proofs of the real number mean value theorem of single variable calculus of single valued functions, I realized that it is not clear to me what is actually meant by mean value. I understand the proofs. I just don't know exactly what the name of the theorem means.
For the case of $f\left[x\right]$ continuous on $x\in\left[x_0,x_1\right]$ and differentiable in $x\in\left(x_0,x_1\right),$ and $c\in\left(x_0,x_1\right)$ such that
$$f^\prime\left[c\right]=\frac{f\left[x_1\right]-f\left[x_0\right]}{x_1-x_0},$$
there are three possible values which might be designated the mean value. Either $c$, $f\left[c\right]$ or $f^\prime\left[c\right].$ Which one is it? Are there good names for the other values in this context?
Best Answer
The mean value would be $\,f'(c)\,$.
This makes a lot more sense if you consider the integral form of MVT $\,f(c) = \frac {1}{b-a}\int _{a}^{b}f(x)\,dx\,$, where the RHS is indeed the "mean value" of $\,f(x)\,$ over $\,[a,b]\,$.