What's under the sqrt() sign is >=0
Here,
f(x)=sqrt(x-3x^2)
Therefore,
x-3x^2>=0
x(1-3x)>=0
Let g(x)=x(1-3x)
Build the sign chart
color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaaa)0color(white)(aaaaaaa)1/3color(white)(aaaaaa)+oo
color(white)(aaaa)xcolor(white)(aaaaaaaa)-color(white)(aaa)0color(white)(aaa)+color(white)(aaaaaa)+
color(white)(aaaa)1-3xcolor(white)(aaaa)+color(white)(aaa)#color(white)(aaaa)+#color(white)(aa)0color(white)(aaaa)-
color(white)(aaaa)g(x)color(white)(aaaaaa)-color(white)(aaa)0color(white)(aaa)+color(white)(aa)0color(white)(aaaa)-
Therefore,
g(x)<=0, =>, x in [0,1/3]
graph{sqrt(x-3x^2) [-0.589, 1.096, -0.307, 0.536]}
