Answer
$c = 4.0$
Work Step by Step
We can find the work done on the particle by the force from $x=0$ to $x = 3.00~m$:
$K_i+W=K_f$
$W = K_f-K_i$
$W = 11.0~J-20.0~J$
$W = -9.0~J$
We can find $c$:
$W = \int^{3.00}_{0}(cx-3.00x^2)~dx$
$W = (\frac{c}{2}x^2-x^3)~\Big\vert^{3.00}_{0}$
$W = \frac{c}{2}(3.00)^2-(3.00)^3-[\frac{c}{2}(0)^2-(0)^3]$
$W = 4.5~c-27.0-0 = -9.0$
$4.5~c = 18.0$
$c = \frac{18.0}{4.5}$
$c = 4.0$
