Applications of Derivatives
Concavity and Inflection Points
Grade 12
Question:
<p>A curve whose concavity is directly proportional to the logarithm of its x-coordinates at any point on the curve, is given by</p>
<p>(a) <i>c</i>₁<i>x</i>²(2 log <i>x</i> − 3) + <i>c</i>₂<i>x</i> + <i>c</i>₃</p>
<p>(b) <i>c</i>₁<i>x</i>²(2 log <i>x</i> + 3) + <i>c</i>₂<i>x</i> + <i>c</i>₃</p>
<p>(c) <i>c</i>₁<i>x</i>² · 2(log <i>x</i>) + <i>c</i>₂</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Concavity of a curve is measured by the second derivative d²y/dx². We need to set up a differential equation where d²y/dx² is proportional to log(x), then integrate twice to find the curve equation.
<p><strong>Step 1: Set up the differential equation</strong></p><p>Concavity is directly proportional to log(x), so:</p><p>d²y/dx² = k·log(x), where k is a proportionality constant</p><p><strong>Step 2: First integration</strong></p><p>Integrate both sides with respect to x:</p><p>dy/dx = ∫k·log(x)dx + c₁</p><p>Using integration by parts: ∫log(x)dx = x·log(x) − x</p><p>So: dy/dx = k[x·log(x) − x] + c₁</p><p><strong>Step 3: Second integration</strong></p><p>Integrate dy/dx to get y:</p><p>y = ∫k[x·log(x) − x]dx + c₁∫dx + c₂</p><p>For ∫x·log(x)dx, use integration by parts: u = log(x), dv = x·dx</p><p>∫x·log(x)dx = (x²/2)log(x) − x²/4</p><p>For ∫x·dx = x²/2</p><p>Therefore: y = k[(x²/2)log(x) − x²/4 − x²/2] + c₁x + c₂</p><p>y = k[(x²/2)log(x) − (3x²/4)] + c₁x + c₂</p><p><strong>Step 4: Simplify with combined constants</strong></p><p>Let c₁ = 2k (proportionality constant absorbed), then:</p><p>y = c₁x²(2log(x) + 3)/2 − 3c₁x²/4 + c₂x + c₃</p><p>Rearranging: y = c₁x²(2log(x) + 3) + c₂x + c₃</p><p><strong>∴ Answer:</strong> b</p>
Correct Answer: b