Question

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

Answer 1

`c = 4/e`

Explanation:

The mean value theorem states that there are numbers `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!