For the interval \(x<-2.568485\):
Let's pick \(x=-3\). Then, \({(-3)}^{3}-6{(-3)}^{2}-15\times -3+18<0\).After simplifying, we get \(-18<0\), which is
true
.
Keep this interval.
.
For the interval \(-2.568485<x<0.915754\):
Let's pick \(x=0\). Then, \({0}^{3}-6\times {0}^{2}-15\times 0+18<0\).After simplifying, we get \(18<0\), which is
false
.
Drop this interval.
.
For the interval \(0.915754<x<7.652731\):
Let's pick \(x=1\). Then, \({1}^{3}-6\times {1}^{2}-15\times 1+18<0\).After simplifying, we get \(-2<0\), which is
true
.
Keep this interval.
.
For the interval \(x>7.652731\):
Let's pick \(x=8\). Then, \({8}^{3}-6\times {8}^{2}-15\times 8+18<0\).After simplifying, we get \(26<0\), which is
false
.
Drop this interval.
.
