91Ó°ÊÓ

The given function is:

Equate the function to zero and solve the equation forx.

So zeros of the function occur at$\mathrm{x}=0,6,2$.

Form interval from zeros of the function.

So intervals are$\left(-∞,-2\right),(-2,0),(0,6),(6,∞)$.

Now take $x=-3$from the intervalrole="math" localid="1654112469416" $\left(-∞,-2\right)$and find the function value.

$$\begin{gathered} \mathrm{f}(-3)={(-3)}^3-4{(-3)}^2-12(-3) \\ \mathrm{f}(-3)=-27 \end{gathered}$$

Hence the function is negative in the interval$\left(-∞,-2\right)$.

Now take $x=-1$from the interval $(-2,0)$and find the function value.

$$\begin{gathered} \mathrm{f}(-1)={(-1)}^3-4{(-1)}^2-12(-1) \\ \mathrm{f}(-1)=7 \end{gathered}$$

Hence the function is positive in the interval$(-2,0)$.

Now take $x=1$from the interval $(0,6)$and find the function value.

$$\begin{gathered} \mathrm{f}\left(1\right)={\left(1\right)}^3-4{\left(1\right)}^2-12\left(1\right) \\ \mathrm{f}\left(1\right)=-15 \end{gathered}$$

Hence the function is negative in the interval$(0,6)$.

Now take $x=7$from the interval $(6,∞)$and find the function value.

$$\begin{gathered} \mathrm{f}\left(7\right)={\left(7\right)}^3-4{\left(7\right)}^2-12\left(7\right) \\ \mathrm{f}\left(7\right)=63 \end{gathered}$$

Hence the function is positive in the interval$(6,∞)$.