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