Answer
$$f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{5}{3}{x^2} - \frac{5}{{90}};\,\,\,\,\,\,\,\left[ { - 1,1} \right] \cr
& {\text{The function f is a probability density function of a random variable X in the interval }} \cr
& \left[ {a,b} \right]{\text{ if}} \cr
& 1{\text{ condition}}:f\left( x \right) \geqslant 0{\text{ for all }}x{\text{ in the interval }}\left[ {a,b} \right].{\text{ then}} \cr
& \frac{5}{3}{x^2} - \frac{5}{{90}} \geqslant 0 \cr
& 150{x^2} - 5 \geqslant 0 \cr
& 30{x^2} - 1 \geqslant 0 \cr
& 30{x^2} \geqslant 1 \cr
& {x^2} \geqslant \frac{1}{{30}} \cr
& - \frac{1}{{30}} \leqslant x \leqslant \frac{1}{{30}} \cr
& \left[ { - \frac{1}{{30}},\frac{1}{{30}}} \right] \cr
& \cr
& \left[ { - 1,1} \right]{\text{ is out of the interval }}\left[ { - \frac{1}{{30}},\frac{1}{{30}}} \right]{\text{ for }}\left[ { - 1, - \frac{1}{{30}}} \right]{\text{ and }}\left[ {\frac{1}{{30}},1} \right] \cr
& f\left( x \right){\text{ is not }} \geqslant {\text{0 for all the interval }}\left[ { - 1,1} \right] \cr
& {\text{Condition 1 is not satisfied}}{\text{, so }} \cr
& \cr
& f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr} $$
