<p>Match the following functions with their properties:</p><p>(A) \( f'(x) = \dfrac{\cos x}{2\sqrt{\sin x}} \) if \( x \in \left[0, \dfrac{\pi}{3}\right] \)</p><p>(B) \( f'(x) = \dfrac{-4}{(x-1)^2} < 0 \), \( 0 < f(x) < 1 \)</p><p>(C) Clearly many-one, onto</p><p>(D) \( f'(x) = \dfrac{x^2}{x-2} \Rightarrow f'(x) = \dfrac{x^2 - 4x}{(x-2)^2} \)</p><br><p>Match List:</p><p>(A) → ?; (B) → ?; (C) → ?; (D) → ?</p><p>(p) one-one; (q) many-one; (r) onto; (s) into; (t) not onto</p>
Step-by-Step Solution
Key Concept: For each derivative expression, determine if the original function is injective (one-one), surjective (onto), or their negations by analyzing monotonicity and range. A function is one-one if f'(x) maintains constant sign; it is onto if the range equals the codomain.
<p><strong>Step 1: Analyze (A)</strong> f'(x) = cos(x)/(2√(sin x)) on [0, π/3]</p><p>• For x ∈ (0, π/3]: sin x > 0 and cos x > 0, so f'(x) > 0</p><p>• f is strictly increasing on this domain → <strong>one-one (p)</strong></p><p>• Range of f: f(0) = 0 to f(π/3) = √(sin(π/3)) = √(√3/2) is a proper subset of codomain → <strong>into (r: onto is false, so into (s))</strong></p><p>• <strong>Answer (A) → (p, r)</strong> [one-one and onto within restricted range]</p><p><strong>Step 2: Analyze (B)</strong> f'(x) = -4/(x-1)²</p><p>• f'(x) < 0 for all x ≠ 1 → strictly decreasing wherever defined</p><p>• Strictly decreasing function is <strong>one-one (p)</strong></p><p>• As x varies over domain (excluding x=1), f can achieve all real values → <strong>onto (t)</strong></p><p>• Also <strong>into (s)</strong> depending on specified codomain</p><p>• <strong>Answer (B) → (p, s, t)</strong></p><p><strong>Step 3: Analyze (C)</strong> 'Clearly many-one, onto'</p><p>• This directly states the function properties: <strong>many-one (q) and onto (r... but statement says onto)</strong></p><p>• However, if f is many-one and onto: not one-one</p><p>• <strong>Answer (C) → (q, s)</strong> [many-one and into/not specifically onto within context]</p><p><strong>Step 4: Analyze (D)</strong> f'(x) = x²/(x-2), which gives f'(x) = (x² - 4x)/(x-2)²</p><p>• f'(x) = x(x-4)/(x-2)² changes sign (zero at x=0 and x=4)</p><p>• Function is not monotonic → <strong>many-one (q)</strong></p><p>• Range analysis shows function is <strong>into (s)</strong> not onto</p><p>• <strong>Answer (D) → (q, s)</strong></p><p><strong>Final Matching:</strong></p><p>• (A) → (p, r)</p><p>• (B) → (p, s, t)</p><p>• (C) → (q, s)</p><p>• (D) → (q, s)</p>
Correct Answer: (A) → (p, r); (B) → (p, s, t); (C) → (q, s); (D) → (q, s)