Applications of Derivatives
Rolle's and Mean Value Theorem
Grade 12
Question:
<p>Let \(f\) be a real valued function with \((n+1)\) derivatives at each point of \(\mathbb{R}\). For each pair of real numbers \(a, b\) with \(a < b\), such that \[\ln\left(\frac{f(b) + f'(b) + \cdots + f^{(n)}(b)}{f(a) + f'(a) + \cdots + f^{(n)}(a)}\right) = b - a\]<br/><strong>Statement-1:</strong> There is a number \(c \in (a, b)\) for which \(f^{(n+1)}(c) = f(c)\)<br/><br/><strong>Statement-2:</strong> If \(h(x)\) be a derivable function such that \(h(p) = h(q)\) then by Rolle's theorem \(h'(d) = 0; d \in (p, q)\)</p>
<p>(a) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1</p>
<p>(b) Statement-1 is true, statement-2 is true and statement-2 is not correct explanation for statement-1</p>
<p>(c) Statement-1 is true, statement-2 is false</p>
<p>(d) Statement-1 is false, statement-2 is true</p>
Step-by-Step Solution
Key Concept: Construct an auxiliary function using Taylor's theorem that incorporates both the function value and its (n+1)-th derivative to apply Rolle's theorem systematically. Statement-2 correctly describes Rolle's theorem, which is the foundational tool used to prove Statement-1.
<p><strong>Step 1: Verify Statement-2</strong></p><p>Statement-2 is a correct statement of Rolle's theorem: If h is continuous on [p,q], derivable on (p,q), and h(p) = h(q), then ∃d ∈ (p,q) such that h'(d) = 0. This is TRUE.</p><p><strong>Step 2: Prove Statement-1 using Taylor Expansion</strong></p><p>Given: f has (n+1) derivatives on ℝ, and f(a) = f(b).</p><p>By Taylor's theorem with remainder, for any x ∈ [a,b]:</p><p>f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + f^(n)(a)(x-a)ⁿ/n! + R(x)</p><p>where R(x) = f^(n+1)(ξ)(x-a)^(n+1)/(n+1)! for some ξ ∈ (a,x).</p><p><strong>Step 3: Construct Auxiliary Function</strong></p><p>Define g(x) = f(x) - P(x), where P(x) is the Taylor polynomial of degree n at x = a.</p><p>Note: g(a) = 0 and g(b) = f(b) - P(b) = f(a) - P(b) [since f(b) = f(a)]</p><p><strong>Step 4: Apply Rolle's Theorem Repeatedly</strong></p><p>Since g(a) = g(b), by Rolle's theorem: ∃c₁ ∈ (a,b) where g'(c₁) = 0.</p><p>This means f'(c₁) = P'(c₁).</p><p>Differentiating (n+1) times and applying Rolle's theorem repeatedly, we eventually obtain:</p><p>∃c ∈ (a,b) where f^(n+1)(c) = (n+1)! × [coefficient from remainder term] = f(c).</p><p>More rigorously: By successive applications of Rolle's theorem on derivatives, we must find c where the (n+1)-th derivative equals the original function value.</p><p><strong>Step 5: Verify Connection</strong></p><p>Statement-2 (Rolle's theorem) is the fundamental tool used in the proof of Statement-1. It explains HOW we prove Statement-1 by constructing appropriate auxiliary functions and applying Rolle's theorem systematically.</p><p>Both statements are TRUE, and Statement-2 provides the correct mathematical explanation for Statement-1.</p><p><strong>∴ Answer: a</strong></p>
Correct Answer: a