Determine where the function $f(x)=2x^{3}-9x^{2}+12x-5$ is increasing and where it is decreasing.

Solution The function $f$ is a polynomial so $f$ is continuous and differentiable at every real number. We find $f^\prime .$ $$f^\prime (x) =6x^{2}-18x+12=6({x-2})({x-1})$$

The Increasing/Decreasing Function Test states that $f$ is increasing on intervals where $f^\prime (x) >0$ and that $f$ is decreasing on intervals where $f^\prime (x) <0$. We solve these inequalities by using the numbers $1$ and $2$ to form three intervals, as shown in Figure 27. Then we determine the sign of $f^\prime (x)$ on each interval, as shown in Table 1.

Figure 27

We conclude that $f$ is increasing on the intervals $(-\infty ,1)$ and $(2,\infty)$, and $f$ is decreasing on the interval $(1,2)$.