Question

Verify the hypothesis and conclusion of Lagrange’s mean value theorem for the function f(x) = 1/(4x - 1), 1 ≤ x ≤ 4

Answer 1

Given as f(x) = 1/(4x - 1), on [1,4]

Here, 4x - 1 > 0

f'(x) has the unique values for all x except 1/4

So, f(x) is continuous in [1,4]

f(x) = 1/(4x - 1)

Differentiate with respect to x

f'(x) = (-1)(4x - 1)^-2(4)

f'(x) = - 4/(4x - 1)²

Where, 4x - 1 > 0

f'(x) has the unique values for all x except 1/4

So, f(x) is differentiable in (1,4)

Therefore both the necessary conditions of lagrange's mean value theorem is satisfied. So, there exist a point c ∈ (1,4)

Differentiate with respect to x

f'(x) = -4/(4x - 1)²

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

f'(c) = -4/(4c - 1)²

For the f'(4), put the value of x = 4 in f'(x)

For the f'(1), put the value of x = 1 in f'(x)

Thus, lagrange's mean value theorem is verified.