Example 1. Find the intervals of increase and decrease of the function

$$
f(x)=x^{3}-\frac{3}{2} x^{2}-6 x+4
$$
Solution. We find the first derivative:

$$
f^{\prime}(x)=3 x^{2}-3 x-6=3\left(x^{2}-x-2\right)
$$

The derivative is zero when $x^{2}-x-2=0$, from which $x_{1}=-1, x_{2}=2$. These points divide the infinite interval into three intervals: $(-\infty,-1),(-1,2),(2, \infty)$. We investigate the sign of the derivative in each of these intervals.

If $x< -1$, then $f^{\prime}(x)>0$, since $(x+1)<0$, $(x-2)<0$, so the function is increasing in the interval $(-\infty, -1)$.

If $-1 < x < 2$, then $f^{\prime}(x)<0$, since $(x+1)>0$, $(x-2)<0$, so the function is decreasing in the interval $(-1,2)$.

Similarly, we verify that the function is increasing in the interval $(2, \infty)$.

Noting that $\lim _{x \rightarrow-\infty} f(x)=-\infty, \lim _{x \rightarrow+\infty} f(x)=+\infty$, we sketch the graph (Fig. 3.6).

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-119.jpg?height=632&width=503&top_left_y=547&top_left_x=286)

Fig. 3.6