Answer
$$f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{3}{{13}}{x^2} - \frac{{12}}{{13}}x + \frac{{45}}{{52}};\,\,\,\,\,\,\,\left[ {0,4} \right] \cr
& {\text{The function f is a probability density function of a random variable X in the interval }} \cr
& \left[ {a,b} \right]{\text{ if}} \cr
& 1{\text{ condition}}:f\left( x \right) \geqslant 0{\text{ for all }}x{\text{ in the interval }}\left[ {a,b} \right].{\text{ then}} \cr
& \frac{3}{{13}}{x^2} - \frac{{12}}{{13}}x + \frac{{45}}{{52}} \geqslant 0 \cr
& 52\left( {\frac{3}{{13}}} \right){x^2} - 52\left( {\frac{{12}}{{13}}x} \right) + 52\left( {\frac{{45}}{{52}}} \right) \geqslant 0 \cr
& 12{x^2} - 48x + 45 \geqslant 0 \cr
& 3\left( {4{x^2} - 16x + 15} \right) \geqslant 0 \cr
& \left( {2x - 5} \right)\left( {2x - 3} \right) \geqslant 0 \cr
& x \geqslant \frac{5}{2}{\text{ and }} \geqslant \frac{3}{2},\,\,\,\,x \leqslant \frac{5}{2}{\text{ and }}x \leqslant \frac{3}{2} \cr
& {\text{then the function is }} \geqslant {\text{0 for the interval}} \cr
& \frac{3}{2} \leqslant x \leqslant \frac{5}{2} \cr
& or \cr
& \left[ {\frac{3}{2},\frac{5}{2}} \right] \cr
& \cr
& 30{x^2} - 1 \geqslant 0 \cr
& 30{x^2} \geqslant 1 \cr
& {x^2} \geqslant \frac{1}{{30}} \cr
& - \frac{1}{{30}} \leqslant x \leqslant \frac{1}{{30}} \cr
& \left[ {0,4} \right] \cr
& \left[ {0,4} \right]{\text{ is out of the interval }}\left[ {0,4} \right]{\text{ for some values}} \cr
& f\left( x \right){\text{ is not }} \geqslant {\text{0 for all the interval }}\left[ {0,4} \right] \cr
& {\text{Condition 1 is not satisfied}}{\text{, so}} \cr
& \cr
& f\left( x \right){\text{ }}is{\text{ }}not{\text{ }}a{\text{ }}probability{\text{ }}density{\text{ }}function \cr} $$
