Question

Verify Lagrange's Mean value Theorem for the functions $f\left(x\right)=e^x$ in $[0,1]$.

Answer 1

$f\left(x\right)=e^x$ in $[0,1]$

function $e^x$ is differential and continuous in interval $(0,1)$

There exsts $C$ such that

$f^1\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$e^c=\frac{e^1-e^o}{1-0}=e-1$

$c=log(e-1)$ which is betwen $(0,1)$

Hence, lang-ranges theorem is verified