Find all numbers C that satisfy the conclusion of the MVT of f(x) = Xln(x) [1,2] ?

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Answer 1 · 2018-05-27T03:34:24

$c = \frac{4}{e}$ $c = 4/e$

The mean value theorem states that there are numbers $c$ $c$ where

$f'(c) = (f(b) - f(a))/(b - a)$ if a function is continuous on $[a, b]$ and differentiable on $(a, b)$ .

Let's do the math.

$f'(c) = (2ln(2) - 0)/(2 - 1)$

$f'(c) = 2ln2$

We now set this equal to the derivative of $f(x)$ to solve for $c$ .

$f'(c) = lnc + c(1/c) = lnc + 1$

Therefore

$lnc + 1 = 2ln2$

$lnc = 2ln2 - 1$

$lnc = ln4 - lne$

$lnc = ln(4/e)$

$c = 4/e$

Hopefully this helps!