Question

Verify Lagrange’ mean value theorem for the functions on the indicated intervals. Find a point ‘c’ in the indicated interval as stated by the Lagrange’ mean value theorem:

f(x) = sin x – sin 2x – x on [0, π]

Answer 1

Given as f(x) = sin x – sin 2x – x on [0, π]

Sin x and cos x functions are continuous everywhere on (−∞, ∞) and differentiable for all arguments. Therefore both the necessary conditions of Lagrange’* mean value theorem is satisfied. So, there exist a point c ∈ (0, π) such that:

Differentiate with respect to x

For the f'(c), put the value of x = c in f'(x)

f'(c) = cos c - 2cos 2c - 1

For the f(π), put the value of x = π in f'(x)

For the f'(0), put the value of x = 0 in f'(x)

From the quadratic equation, ax2 + bx + c = 0

Thus, lagrange'* mean value theorem is verified.