Answer
$$\eqalign{
& {\text{Inflection point }}\left( {0,0} \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( { - \infty ,0} \right) \cr
& {\text{Concave downward}}:{\text{ }}\left( {0,\infty } \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 3x - 5{x^3} \cr
& {\text{Calculate the second derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {3x - 5{x^3}} \right] \cr
& f'\left( x \right) = 3 - 15{x^2} \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {3 - 15{x^2}} \right] \cr
& f''\left( x \right) = - 30x \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& - 30x = 0 \cr
& x = 0 \cr
& {\text{Set the intervals }}\left( { - \infty ,0} \right),\left( {0,\infty } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( { - \infty ,0} \right)}&{\left( {0,\infty } \right)} \\
{{\text{Test Value}}}&{x = - 1}&{x = 1} \\
{{\text{Sign of }}f''\left( x \right)}&{f''\left( { - 1} \right) = 30 > 0}&{f''\left( 1 \right) = - 30 < 0} \\
{{\text{Conclusion}}}&{{\text{Concave upward}}}&{{\text{Concave downward}}}
\end{array}}\]
$$\eqalign{
& {\text{The inflection point occurs at }}x = 0 \cr
& f\left( 0 \right) = 3\left( 0 \right) - 5{\left( 0 \right)^3} \cr
& f\left( 0 \right) = 0 \cr
& {\text{Inflection point }}\left( {0,0} \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( { - \infty ,0} \right) \cr
& {\text{Concave downward}}:{\text{ }}\left( {0,\infty } \right) \cr} $$
