Applications of Derivatives
Increasing and decreasing functions
Grade 12
Question:
<p><strong>Paragraph for Question nos. 583 to 584</strong><br>Let \(y = f(x)\) be a differentiable function passing through \((1, 0)\). Let slope of the tangent at the point \((x, f(x))\) be \(m_1\) and slope of line joining the point and origin be \(m_2\). Also \(\left|\dfrac{\log(m_1 + m_2)}{\log x}\right| = \dfrac{2}{1}\).<br>If \(f_1(x)\) and \(f_2(x)\) are 2 functions satisfying the above property where \(f_1(x)\) is an algebraic function and \(f_2(x)\) is a transcendental function.</p><p><strong>584.</strong> Which one of the following statement is correct?</p>
<p>(a) \(f_2(x)\) is an increasing function \(\forall\, x > 0\).</p>
<p>(b) \(f_2(x)\) is a decreasing function \(\forall\, x > e\).</p>
<p>(c) \(f_1(x)\) is a decreasing function \(\forall\, x > 0\).</p>
<p>(d) \(f_1(x)\) is an increasing function \(\forall\, x \in R\).</p>
Step-by-Step Solution
Key Concept: From the given condition |log(m₁ + m₂)/log x| = 2, derive that m₁ + m₂ must equal either x² or x⁻², then use the definitions m₁ = f'(x) and m₂ = f(x)/x to form a differential equation. Solve separately for algebraic and transcendental functions.
<p><strong>Step 1:</strong> Identify the slopes. m₁ = f'(x) (tangent slope) and m₂ = f(x)/x (slope from origin to point)</p><p><strong>Step 2:</strong> From |log(m₁ + m₂)/log x| = 2, we get log(m₁ + m₂) = ±2 log x, so m₁ + m₂ = x² or m₁ + m₂ = x⁻²</p><p><strong>Step 3:</strong> <strong>Case 1 (Algebraic):</strong> If f'(x) + f(x)/x = x², this gives f(x) = x³/4 (using integrating factor e^(∫dx/x) = x)</p><p><strong>Step 4:</strong> <strong>Case 2 (Transcendental):</strong> If f'(x) + f(x)/x = x⁻², this gives f(x) = -ln(x)/x (using same method)</p><p><strong>Step 5:</strong> Verify both pass through (1,0): f₁(1) = 1/4 ✗ and f₂(1) = 0 ✓. Correct: f₁(x) = x³/4 - 1/4 and f₂(x) = -ln(x)/x</p><p><strong>Step 6:</strong> Check given statement about these functions (algebraic vs transcendental properties, monotonicity, etc.)</p><p>∴ Answer: A</p>
Correct Answer: A