Answer
$$
y=6x-9
$$
(a)
The critical numbers : There are no critical numbers.
(b)
The function is increasing on interval $(-\infty, \infty )$ .
(c)
No interval that the function is decreasing , since $ f^{\prime}(x)$ is positive for all $x$.
Work Step by Step
$$
y=6x-9
$$
First, find the points where the derivative $f^{\prime }$ is $0$.
Here
$$
\begin{aligned}
f^{\prime}(x) &=6 \\
\end{aligned}
$$
Solve the equation $f^{\prime}(x) =0 $ to get
$$
\begin{aligned}
& f^{\prime}(x) =6 \ne 0\\
\Rightarrow\quad\quad\quad\quad\quad\quad\quad\quad\quad\\
&\text{ [Then, there are no critical numbers ]}
\end{aligned}
$$
(a)
The critical numbers : There are no critical numbers.
Now, we can use the first derivative test.
Check the sign of $f^{\prime}(x)$.
$$
\begin{aligned}
f^{\prime}(x) &=6 \gt 0 , \text {for all } x\\
\end{aligned}
$$
WE see that $ f^{\prime}(x)$ is positive for all $x$, so $f(x)$ is increasing on $(-\infty, \infty )$
So,
(b)
The function is increasing on interval $(-\infty, \infty )$ .
(c)
No interval that the function is decreasing , since $ f^{\prime}(x)$ is positive for all $x$.
