Limits, Continuity & Differentiability
Differentiability and derivatives
Grade 12
Question:
<p>If \(g(x) = (x^2 + 2x + 3)f(x)\), \(f(0) = 5\) and \(\displaystyle\lim_{x \to 0}\left(\frac{f(x) - 5}{x}\right) = 4\), then \(g'(0)\) is equal to:</p>
<p>(a) 22</p>
<p>(b) 18</p>
<p>(c) 23</p>
<p>(d) 25</p>
Step-by-Step Solution
Key Concept: Recognize that the given limit defines f'(0) = 4, then apply the product rule to g(x) = (x² + 2x + 3)f(x) at x = 0. The polynomial factor and its derivative at x = 0 are both easily computable.
<p><strong>Step 1:</strong> Identify f'(0) from the given limit.</p><p>Given: $\lim_{x \to 0}\frac{f(x) - 5}{x} = 4$ and $f(0) = 5$</p><p>This limit is the definition of $f'(0) = 4$</p><p><strong>Step 2:</strong> Apply the product rule to $g(x) = (x^2 + 2x + 3)f(x)$.</p><p>$$g'(x) = (2x + 2)f(x) + (x^2 + 2x + 3)f'(x)$$</p><p><strong>Step 3:</strong> Evaluate $g'(0)$ using $f(0) = 5$ and $f'(0) = 4$.</p><p>$$g'(0) = (2(0) + 2)f(0) + (0^2 + 2(0) + 3)f'(0)$$</p><p>$$g'(0) = (2)(5) + (3)(4)$$</p><p>$$g'(0) = 10 + 12 = 22$$</p><p>∴ Answer: C</p>
Correct Answer: C