Question

If f and $g$ are the functions whose graphs are shown, let $P(x)=f(x)g(x),Q(x)=\frac{f(x)}{g(x)}$ and $C(x)=f(g(x))$. The value of $\left(P^{\prime }(2)-C^{\prime }(2)\right)Q^{\prime }(2)$ equals

• A. $3/2$
• B. 3
• C. -3
• D. -6

Answer 1

Answer: (B)

$P^{\prime }(x)=f(x)g^{\prime }(x)+g(x)f^{\prime }(x)$
$P^{\prime }(2)f(2)g^{\prime }(2)+g(2)f^{\prime }(2)$
$=(1)(2)+4(-1)$
$=-2$
$Q^{\prime }(x)=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{g^2(x)}$
$Q^{\prime }(2)=\frac{(4)(-1)-(1)(2)}{16}=-\frac{6}{16}=-\frac{3}{8}$
$C^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x)$
$C^{\prime }(2)=f^{\prime }(4)\cdot 2=3\cdot 2=6$