a) Determine whether the Mean Value Theorem applies to " f(x) = ln(2x) " on the interval [1,e]

b) If so, determine the points that are guaranteed to exist by the MVT

Question

a) Determine whether the Mean Value Theorem applies to " f(x) = ln(2x) " on the interval [1,e]

b) If so, determine the points that are guaranteed to exist by the MVT

anonymous · Answer

$f(x)=\ln(2x)$ must be continuous on $[1,e]$ and differentiable on $(1,e)$. (You can verify that this is indeed the case.)

You then want to find the values of $c$ such that
$$f'(c)=\frac{f(b)-f(a)}{b-a}$$
where $a=1$ and $b=e$. The derivative of $\ln(2x)$ is $f'(x)=\dfrac{1}{x}$, so you have to find $c$ that satisfies the equation,
$$\frac{1}{c}=\frac{\ln(2e)-\ln2}{e-1}$$