Question

Verify Langrange’s mean value theorem for the function: f(x) = x(1 – log x) and find the value of ‘c’ in the interval [1, 2]

Answer 1

Given function ‘f’ is continuous in [1, 2] and differentiable in (1, 2) 

f(x) = x(1 – log x) = x – x log x 

f'(x) = 1 – x × (1/x) – log x = 1 – 1 – log x

f'(x) = – log x

According to Langrange’s Mean Value Theorem, E a real number c ∈ (1, 2) s.t.,