The functions $\mathrm{p}$ and $\mathrm{q}$ are continuous for $1\leq x\leq5$. The functi

Question

The functions $\mathrm{p}$ and $\mathrm{q}$ are continuous for $1\leq x\leq5$. The function $\mathrm{p}$ is strictly increasing, while $\mathrm{q}$ is strictly decreasing and $\mathrm{q}(x)>0$. Selected values are shown below. $x$$1$$2$$3$$4$$5$ $\mathrm{p}(x)$$-10$$-8$$-6$$-4$$5$ $\mathrm{q}(x)$$5$$4$$3$$2$$1$ Using the information in the table, Find $\mathrm{p}\mathrm{q}(2)$. [1] Solve $\mathrm{p}^{-1}\mathrm{q}(x)=5$. [2] The function $\mathrm{r}$ is defined by $\mathrm{r}(x)=\frac{\mathrm{p}(x)}{\mathrm{q}(x)}$. It is further given that $\mathrm{p}'(3)=3$ and $\mathrm{q}'(3)=-1$. Explain why $\mathrm{r}^{-1}$ does not exist on the given domain.[2] Find the equation of the normal to $y=\mathrm{r}(x)$ at $x=3$.[3]

Accepted Answer

(a) $-4$ (b) $x=1$ (c) $\mathrm{r}^{-1}$ does not exist on the given domain (d) $y+2=-3(x-3)$

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(\mathrm{p}(x)\)	\(-10\)	\(-8\)	\(-6\)	\(-4\)	\(5\)
\(\mathrm{q}(x)\)	\(5\)	\(4\)	\(3\)	\(2\)	\(1\)

Normal from tabulated function values

Normal from tabulated function values

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