Question

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

• A. 4
• B. -4
• C. 8
• D. -8

Answer 1

Answer: (D)

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