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: Use the limit definition directly:
$g'(0) = \lim_{x \to 0}\frac{g(x) - g(0)}{x}$, with $g(0) = 5$.

Step 2: Expand:
$\frac{(x + 1)^2 f(x) - 5}{x} = (x + 1)^2\frac{f(x) - 5}{x} + \frac{5[(x + 1)^2 - 1]}{x} \to 4 + 5 \times 2 = 14$.

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