<p>If \(f(x) = x^4 \tan x^3 - x\ln(1 + x^2)\), then the value of \(\dfrac{d^4 f(x)}{dx^4}\) at \(x = 0\) is:</p>
Step-by-Step Solution
Key Concept: Use Taylor series expansion of f(x) around x=0; the coefficient of x^4 in the expansion equals f⁽⁴⁾(0)/4!, so f⁽⁴⁾(0) = 24 times the x^4 coefficient.
<p><strong>Step 1:</strong> Expand each term using Taylor series around x=0.</p><p>For tan(x³): tan(u) = u + u³/3 + ..., so tan(x³) = x³ + x⁹/3 + ...</p><p>Thus: x⁴·tan(x³) = x⁴(x³ + x⁹/3 + ...) = x⁷ + x¹³/3 + ... (no x⁴ term)</p><p><strong>Step 2:</strong> Expand ln(1 + x²).</p><p>ln(1 + u) = u - u²/2 + u³/3 - ..., so ln(1 + x²) = x² - x⁴/2 + x⁶/3 - ...</p><p>Thus: x·ln(1 + x²) = x(x² - x⁴/2 + ...) = x³ - x⁵/2 + ... (no x⁴ term)</p><p><strong>Step 3:</strong> Combine results.</p><p>f(x) = [x⁷ + ...] - [x³ - x⁵/2 + ...] = -x³ + x⁵/2 + x⁷ + ...</p><p>The coefficient of x⁴ in f(x) is <strong>0</strong>.</p><p><strong>Step 4:</strong> Use Taylor coefficient relation.</p><p>f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + f⁽⁴⁾(0)x⁴/4! + ...</p><p>Coefficient of x⁴ = f⁽⁴⁾(0)/24 = 0</p><p>∴ f⁽⁴⁾(0) = <strong>0</strong></p><p><strong>Answer: A (assuming A = 0)</strong></p>
Correct Answer: A