Answer
$$\eqalign{
& \left( {\text{a}} \right)x = 3 \cr
& \left( {\text{b}} \right){\text{Increasing on: }}\left( { - \infty ,\infty } \right) \cr
& \left( {\text{c}} \right){\text{No relative extrema}} \cr} $$
Work Step by Step
$$\eqalign{
& {\text{We have }}f\left( x \right) = {\left( {x - 3} \right)^{1/3}} \cr
& \left( {\text{a}} \right){\text{Calculating the first derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {x - 3} \right)}^{1/3}}} \right] \cr
& f'\left( x \right) = \frac{1}{3}{\left( {x - 3} \right)^{ - 2/3}} \cr
& f'\left( x \right) = \frac{1}{{3{{\left( {x - 3} \right)}^{2/3}}}} \cr
& {\text{There are no points at which }}f'\left( x \right) = 0 \cr
& {\text{The derivative is not defined at }}x - 3 = 0,{\text{ so we obtain the critical }} \cr
& {\text{point }}x = 3 \cr
& \cr
& \left( {\text{b}} \right) \cr
& {\text{Set the intervals }}\left( { - \infty ,3} \right),\left( {3,\infty } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 180 }}} \right) \cr} $$
\[\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( { - \infty ,3} \right)}&{\left( {3,\infty } \right)} \\
{{\text{Test Value}}}&{x = 0}&{x = 4} \\
{{\text{Sign of }}f'\left( x \right)}&{{\text{ }}f'\left( 0 \right) > 0}&{{\text{ }}f'\left( 4 \right) = \frac{1}{3} > 0} \\
{{\text{Conclusion}}}&{{\text{Increasing}}}&{{\text{Increasing}}}
\end{array}\]
$$\eqalign{
& \cr
& \left( {\text{c}} \right) \cr
& {\text{No relative minimum or relative maximum}} \cr
& \cr
& \left( {\text{a}} \right)x = 3 \cr
& \left( {\text{b}} \right){\text{Increasing on: }}\left( { - \infty ,\infty } \right) \cr
& \left( {\text{c}} \right){\text{No relative extrema}} \cr} $$
