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 = 4/e#

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!