Applications of Derivatives
Rolle's theorem and mean value theorem
Grade 12
Question:
<p><strong>510.</strong> Let \(f(x)\) be a derivable function and \(f(\alpha) = f(\beta) = 0\) \((a < \beta)\), then in the interval \((\alpha, \beta)\):</p>
<p>\(f(x) + f'(x) = 0\) has at least one real root</p>
<p>\(f(x) - f'(x) = 0\) has at least one real root</p>
<p>\(f(x) f'(x) = 0\) has at least one real root</p>
<p>\((f'(x))^2 + f(x)f''(x) = 0\) has at least two real roots</p>
Step-by-Step Solution
Key Concept: By Rolle's Theorem, since f is derivable and f(α) = f(β) = 0, there exists at least one point c ∈ (α,β) where f'(c) = 0. The number of such critical points equals the number of zeros of f'(x) in that interval.
<p><strong>Step 1:</strong> Given that f(x) is derivable on [α, β] and f(α) = f(β) = 0, we can apply Rolle's Theorem.</p><p><strong>Step 2:</strong> Rolle's Theorem states: If a function is continuous on [α, β], derivable on (α, β), and f(α) = f(β), then there exists at least one c ∈ (α, β) such that f'(c) = 0.</p><p><strong>Step 3:</strong> Since f(α) = f(β) = 0, all conditions of Rolle's Theorem are satisfied. Therefore, f'(x) = 0 has at least one solution in the open interval (α, β).</p><p><strong>Step 4:</strong> The minimum number of zeros of f'(x) in (α, β) is <strong>at least 1</strong>. If the question asks for the guaranteed minimum, the answer is 1 (or option indicating existence of at least one zero).</p><p>∴ Answer: A</p>
Correct Answer: A