Question

Verify the Langrange’s theorem for the following:

(a) f(x) = (x – 1) (x – 2) (x – 3), x ∈ [0, 4]

(b) f(x) = \( \begin{cases} 1 + x, & x < 2\\ 5 - x, & x \geq2 \end{cases}\) , x ∈ [1, 3]

Answer 1

(a) Given function

f(x) = (x – 1) (x – 2) (x – 3), x ∈ [0, 4]

⇒ f(x) = x³ – 6x² + 11x – 6, x ∈ [0, 4]

Clearly,f(x) is continuous in interval [0,4] and f'(x) is defined and exists in interval (0,4). 

Hence, function satisfies Lagrange’s mean value theorem.

Hence, langrange's theorem verified.

(b) Given function

f(x) = \( \begin{cases} 1 + x, & x < 2\\ 5 - x, & x \geq2 \end{cases}\) , x ∈ [1, 3]

Hence,f(x) is continuous and differentiable in interval (1, 3) excepting x = 2.

Testing of continuity at x = 2

Function is continuous at x = 2.

Testing of differentiability at x = 2.

So, function f(x) is not differentiable at x = 2.

Here condition for Lagrange’s theorem does not satisfied. 

Hence, Lagrange’s does not verified.