Answer
$$
H(r)=\frac{300}{\left(1+0.03r^{2}\right)}
$$
where $r$ is the mortgage rate (in percent).
we conclude that :
(a) $H(r) $ is increasing on nowhere
(b) $ H^{\prime}(r) \lt 0 $ on $( 0, \infty) $ so $H(r)$ is
decreasing on $( 0, \infty)$.
Work Step by Step
$$
H(r) =\frac{300}{\left(1+0.03r^{2}\right)} =300 \left(1+0.03r^{2}\right)^{-1}
$$
where $r$ is the mortgage rate (in percent).
The first derivative is
$$
\begin{aligned}
H^{\prime}(r) &=300\left[-1\left(1+0.03 r^{2}\right)^{-2}(0.06 r)\right] \\
&=\frac{-18 r}{\left(1+0.03 r^{2}\right)^{2}}
\end{aligned}
$$
Since $r$ is a mortgage rate (in percent), it is always positive.
Note that $H(r)$ is only defined for $r \gt 0 $. Thus, $H^{\prime}(r)$ is always negative.
So, we conclude that :
(a) $H(r) $ is increasing on nowhere
(b) $ H^{\prime}(r) \lt 0 $ on $( 0, \infty) $ so $H(r)$ is
decreasing on $( 0, \infty)$.
