Answer
$$\eqalign{
& {\text{Increasing on }}\left( { - \infty , - \frac{1}{2}} \right),\left( {0,\frac{1}{2}} \right) \cr
& {\text{Decreasing on }}\left( { - \frac{1}{2},0} \right),\,\left( {\frac{1}{2},\infty } \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = - 2{x^4} + {x^2} + 10 \cr
& {\text{Derivative}} \cr
& f'\left( x \right) = - 8{x^3} + 2x \cr
& {\text{Set the derivative to 0}} \cr
& - 8{x^3} + 2x = 0 \cr
& 2x\left( { - 4{x^2} + 1} \right) = 0 \cr
& 2x\left( {1 + 2x} \right)\left( {1 - 2x} \right) = 0 \cr
& {\text{Solving the equation we obtain}} \cr
& x = 0,\,\,\,x = - \frac{1}{2}{\text{ and }}x = \frac{1}{2} \cr
& {\text{From the critical values and the domain }}\left( { - \infty ,\infty } \right){\text{ we have}} \cr
& \left( { - \infty , - \frac{1}{2}} \right),\,\,\left( { - \frac{1}{2},0} \right),\,\,\left( {0,\frac{1}{2}} \right),\,\,\left( {\frac{1}{2},\infty } \right) \cr
& {\text{Now}}{\text{, we will evaluate the critical value and resume in a table}} \cr} $$
\[\begin{array}{*{20}{c}}
{{\rm{Interval}}}&{{\rm{Test\ value }}\left( x \right)}&{{\rm{Sign\ of }}f'\left( x \right)}&{{\rm{Behavior \ of }}f\left( x \right)}\\
{\left( { - \infty , - \frac{1}{2}} \right)}&{ - 1}& + &{{\rm{Increasing}}}\\
{\left( { - \frac{1}{2},0} \right)}&{ - \frac{1}{4}}& - &{{\rm{Decreasing}}}\\
{\left( {0,\frac{1}{2}} \right)}&{\frac{1}{4}}& + &{{\rm{Increasing}}}\\
{\left( {\frac{1}{2},\infty } \right)}&1& - &{{\rm{Decreasing}}}\\
{}&{}&{}&{}\\
{}&{}&{}&{}
\end{array}\]
$$\eqalign{
& {\text{From the table we can conlude that the function is:}} \cr
& {\text{Increasing on }}\left( { - \infty , - \frac{1}{2}} \right),\left( {0,\frac{1}{2}} \right) \cr
& {\text{Decreasing on }}\left( { - \frac{1}{2},0} \right),\,\left( {\frac{1}{2},\infty } \right) \cr} $$
