Question

If \(g(x) = (x^2+2x+1)\cdot f(x)\) such that \(f(0) = 5\) and \(\underset{x\rightarrow 0}{lim}\frac{f(x)-5}{x} = 4\) then \(g^'(0) =\)

• A. \(20\)
• B. \(12\)
• C. \(18\)
• D. \(14\)

Hint:

Use the product rule and the definition of the derivative of f at 0.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(\lim_{x \to 0}\frac{f(x) - f(0)}{x} = f'(0)\), so the given limit equals \(f'(0) = 4\).

Step 2: Key Formula or Approach:
\(g(x) = (x + 1)^2 f(x)\), so \(g'(x) = 2(x + 1)f(x) + (x + 1)^2 f'(x)\).

Step 3: Detailed Explanation:
At \(x = 0\): \(g'(0) = 2(1)f(0) + (1)^2 f'(0)\).
\[ g'(0) = 2 \times 5 + 4 = 14 \]
Option B (\(12\)) would result from forgetting the factor \(2\) in \(2(x + 1)\), and option A (\(20\)) from taking only the first term with \(f(0) \times 4\).

Final Answer:
\(g'(0) = 14\), option (D). \[ \boxed{14} \]