Answer
$$
f(x)=3x^{4}+8 x^{3}-18x^{2}+5
$$
(a)
The critical numbers are: $-3 , 1,$ and $0$.
(b)
The function is increasing on intervals $(-3, 0 ) \text { and } (1, \infty).$
(c)
The function is decreasing on intervals $(-\infty, -3 ) \text { and } (0, 1). $
Work Step by Step
$$
f(x)=3x^{4}+8 x^{3}-18x^{2}+5
$$
First, find the points where the derivative $f^{\prime }$ is $0$.
Here
$$
\begin{aligned}
f^{\prime}(x) &=3 (4)x^{3}+8 (3) x^{2}-18 (2)x \\
&=12x^{3}+24 x^{2}-36x \\
&=12x( x^{2}+2 x-3) \\
\end{aligned}
$$
Solve the equation $f^{\prime}(x) =0 $ to get
$$
\begin{aligned}
f^{\prime}(x) =12x( x^{2}+2 x-3)&=0\\
\Rightarrow\quad\quad\quad\quad\quad\quad\quad\quad\quad\\
f^{\prime}(x)=12x(x+3)(x-1)&=0 \\
\Rightarrow\quad\quad\quad\quad\quad\quad\quad\quad\quad\\
x =1 , \quad x =0, \quad &\text {and} \quad x =-3 \\
\end{aligned}
$$
(a)
The critical numbers are: $-3 , 1,$ and $0$.
Now, we can use the first derivative test.
Check the sign of $f^{\prime}(x)$ in the intervals
$$
(-\infty, -3 ), \quad ( -3, 0), \quad ( 0, 1),\quad \text {and } ( 1, \infty) .
$$
(1)
Test a number in the interval $(-\infty, -3)$ say $-4$:
$$
\begin{aligned}
f^{\prime}(-4) &=12(-4)( (-4)^{2}+2 (-4)-3) \\
&=-240 \\
& \lt 0
\end{aligned}
$$
to see that $ f^{\prime}(x)$ is negative in that interval, so $f(x)$ is decreasing on $(-\infty, -3 )$
(2)
Test a number in the interval $( -3, 0)$ say $-1$:
$$
\begin{aligned}
f^{\prime}(-1) &=12(-1)( (-1)^{2}+2 (-1)-3) \\
&=48\\
& \gt 0
\end{aligned}
$$
to see that $ f^{\prime}(x)$ is positive in that interval, so $f(x)$ is increasing on $( -3 , 0),$
(3)
Test a number in the interval $(0, 1)$ say $\frac{1}{2}$:
$$
\begin{aligned}
f^{\prime}(\frac{1}{2}) &=12(\frac{1}{2})( (\frac{1}{2})^{2}+2 (\frac{1}{2})-3) \\
&=-\frac{21}{2}\\
& \lt 0
\end{aligned}
$$
to see that $ f^{\prime}(x)$ is negative in that interval, so $f(x)$ is decreasing on $(0, 1 ) .$
(4)
Test a number in the interval $(1, \infty)$ say $2$:
$$
\begin{aligned}
f^{\prime}(2) &=12(2)( (2)^{2}+2 (2)-3) \\
&=120 \\
& \gt 0
\end{aligned}
$$
to see that $ f^{\prime}(x)$ is positive in that interval, so $f(x)$ is increasing on $(1, \infty ) .$
So,
(b) The function is increasing on intervals $(-3, 0 ) \text { and } (1, \infty), $
(c) The function is decreasing on intervals $(-\infty, -3 ) \text { and } (0, 1). $
