Answer
$${\text{Increasing on: }}\left( { - \infty ,\infty } \right)$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\left( {x + 1} \right)^3} \cr
& {\text{Calculate the first derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {x + 1} \right)}^3}} \right] \cr
& f'\left( x \right) = 3{\left( {x + 1} \right)^2} \cr
& {\text{Find the critical points, set the first derivative to }}0 \cr
& f'\left( x \right) = 3{\left( {x + 1} \right)^2} \cr
& 3{\left( {x + 1} \right)^2} = 0 \cr
& {\text{We have the critical point }}x = - 1 \cr
& {\text{Set the intervals }}\left( { - \infty , - 1} \right){\text{ and }}\left( { - 1,\infty } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 178 }}} \right) \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( { - \infty , - 1} \right)}&{\left( { - 1,\infty } \right)} \\
{{\text{Test Value}}}&{x = - 2}&{x = 2} \\
{{\text{Sign of }}f'\left( x \right)}&{f'\left( { - 3} \right) = 3 > 0}&{f'\left( 3 \right) = 27 > 0} \\
{{\text{Conclusion}}}&{{\text{Increasing}}}&{{\text{Increasing}}}
\end{array}}\]
$$\eqalign{
& {\text{By Theorem 3}}{\text{.5 }}h{\text{ is:}} \cr
& {\text{Increasing on: }}\left( { - \infty ,\infty } \right) \cr} $$
