Question

Among the following, Rolle's theorem is not applicable for

• A. \( f(x) = x^3 - 4x \) in \([-2, 2]\)
• B. \( f(x) = |x| \) in \([-1, 1]\)
• C. \( f(x) = (x-a)^m (x-b)^n \) in \([a, b], m, n>0\)
• D. \( f(x) = x^2 - 3x + 2 \) in \([1, 2]\)

Hint:

For Rolle's theorem, always check differentiability as well as continuity — absolute value functions like \(|x|\) are non-differentiable at 0.

Answer 1

The Correct Option is B

Rolle's Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \in (a, b) \) such that \( f'(c) = 0 \).

Let's analyze the options:

Option	Analysis
\( f(x) = x^3 - 4x \) in \([-2, 2]\)	1. Continuous on \([-2, 2]\). 2. Differentiable on \((-2, 2)\). 3. \( f(-2) = -8, f(2) = 0 \). Not applicable as \( f(a) \neq f(b) \).
\( f(x) = |x| \) in \([-1, 1]\)	1. Continuous on \([-1, 1]\). 2. Not differentiable at \( x = 0 \) (cusp point). 3. \( f(-1) = 1, f(1) = 1 \). Not applicable due to non-differentiability.
\( f(x) = (x-a)^m (x-b)^n \) in \([a, b]\)	Assuming \( m, n > 0 \): 1. Continuous on \([a, b]\). 2. Differentiable on \((a, b)\). 3. \( f(a) = f(b) = 0 \). Rolle's theorem is applicable.
\( f(x) = x^2 - 3x + 2 \) in \([1, 2]\)	1. Continuous on \([1, 2]\). 2. Differentiable on \((1, 2)\). 3. \( f(1) = 0, f(2) = 0 \). Rolle's theorem is applicable.

Among the given options, Rolle's theorem is not applicable for \( f(x) = |x| \) in \([-1, 1]\) due to its non-differentiability at \( x = 0 \).