Applications of Derivatives
Higher Order Derivatives and Periodicity
Grade 12
Question:
<p>Let \(f(x) = \sin^6\left(\frac{x}{4}\right) + \cos^6\left(\frac{x}{4}\right)\). If \(f_n(x)\) denotes the \(n\)-th derivative of \(f\) evaluated at \(x\), then which of the following hold?</p>
<p>(a) \(f^{(2014)}(0) = -\frac{3}{8}\)</p>
<p>(b) \(f^{(2015)}(0) = \frac{3}{8}\)</p>
<p>(c) \(f^{(2010)}\left(\frac{\pi}{2}\right) = 0\)</p>
<p>(d) \(f^{(2011)}\left(\frac{\pi}{2}\right) = \frac{3}{8}\)</p>
Step-by-Step Solution
Key Concept: Simplify f(x) using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), then express as a trigonometric function with periodic derivatives. The key is recognizing that derivatives of periodic functions repeat with a fixed period.
<p><strong>Step 1: Simplify f(x) using algebraic identity</strong></p><p>Let a = sin²(x/4) and b = cos²(x/4). Then a + b = 1.</p><p>We need a³ + b³ = (a + b)³ - 3ab(a + b) = 1 - 3ab = 1 - 3sin²(x/4)cos²(x/4)</p><p>Since sin²(x/4)cos²(x/4) = ¼sin²(x/2), we have:</p><p>f(x) = 1 - 3/4 · sin²(x/2) = 1 - 3/4 · (1 - cos(x))/2 = 5/8 + (3/8)cos(x)</p><p><strong>Step 2: Find the pattern of derivatives</strong></p><p>f(x) = 5/8 + (3/8)cos(x)</p><p>f'(x) = -(3/8)sin(x)</p><p>f''(x) = -(3/8)cos(x)</p><p>f'''(x) = (3/8)sin(x)</p><p>f⁽⁴⁾(x) = (3/8)cos(x)</p><p>The derivatives repeat with period 4.</p><p><strong>Step 3: Evaluate f⁽²⁰¹⁴⁾(0)</strong></p><p>2014 = 4 × 503 + 2, so f⁽²⁰¹⁴⁾(x) = f''(x) = -(3/8)cos(x)</p><p>f⁽²⁰¹⁴⁾(0) = -(3/8)cos(0) = -3/8 ✓ Option (a) is TRUE</p><p><strong>Step 4: Evaluate f⁽²⁰¹⁵⁾(0)</strong></p><p>2015 = 4 × 503 + 3, so f⁽²⁰¹⁵⁾(x) = f'''(x) = (3/8)sin(x)</p><p>f⁽²⁰¹⁵⁾(0) = (3/8)sin(0) = 0 ✗ Option (b) is FALSE</p><p><strong>Step 5: Evaluate f⁽²⁰¹⁰⁾(π/2)</strong></p><p>2010 = 4 × 502 + 2, so f⁽²⁰¹⁰⁾(x) = f''(x) = -(3/8)cos(x)</p><p>f⁽²⁰¹⁰⁾(π/2) = -(3/8)cos(π/2) = 0 ✓ Option (c) is TRUE</p><p><strong>Step 6: Evaluate f⁽²⁰¹¹⁾(π/2)</strong></p><p>2011 = 4 × 502 + 3, so f⁽²⁰¹¹⁾(x) = f'''(x) = (3/8)sin(x)</p><p>f⁽²⁰¹¹⁾(π/2) = (3/8)sin(π/2) = 3/8 ✓ Option (d) is TRUE</p><p><strong>∴ Answer:</strong> a, c, d</p>
Correct Answer: a, c, d