Question

f(x) = ln x, [1, 8]. Using mean value theorem, find all numbers c that satisfy the conclusion of the Mean Value Theorem.

Answer 1

The mean value theorem states that if f(x)f(x) is continuous on an interval [a,b][a,b] and differentiable on (a,b)(a,b),
then there exists a c∈(a,b)c∈(a,b) 
such thatf′(c)=[f(b)−f(a)]/b−a
The function f(x)=ln⁡(x) is continuous on an interval [1,8] and differentiable on (1,8).
The derivative of ln⁡(x) is 1x in the interval (1,8).
Hence, by mean value theorem, ∃c∈(1,8) such that
f′(c)=1/c
      = [ f(8)−f(1)]/8−1
      = [ ln(8)−ln(1) ]/8−1
      = [3 ln(2)−0]/7
      = 3ln(2)/7
Hence, the desired point c is 7/3 ln(2)