Question

Let $f :(-5,5) \rightarrow R$ be a differentiable function with $f (4)=1, f ^{\prime}(4)=1, f (0)=-1$ and $f ^{\prime}(0)=1$. If $g(x)=\left(f\left(2 f^2(x)+2\right)\right)^2$, then $g^{\prime}(0)$ equals

• A. 4
• B. -4
• C. 8
• D. -8

Answer 1

Answer: (D)

$g^{\prime}(x)=2 f\left(2 f^2(x)+2\right) \times f^{\prime}\left(2 f^2(x)+2\right) \times 4 f(x) \cdot f^{\prime}(x)$
$g^{\prime}(0)=2 f\left(2 f^2(0)+2\right) \times f^{\prime}\left(2 f^2(0)+2\right) \times 4 f(0) \cdot f^{\prime}(0) $
$=2 f(4) \cdot f^{\prime}(4) \cdot 4 f(0) f^{\prime}(0)=2 \times 4(-1)(1)=-8$