Answer:
[tex]f[/tex] is sometimes positive and sometimes negative.
Step-by-step explanation:
[tex]f(x)=(x-3)(x+2)(x+4)(x+4)(x-1)(2x-9)[/tex]
Take [tex]x=-1\in(-2,\frac{9}{2})\ as-2<-1<\frac{9}{2}[/tex]
[tex]f(-1)=(-1-3)(-1+2)(-1+4)(-1-1)(-2-9)\\\\=(-4)(1)(3)(-2)(-11)\\\\=-264\\\\f(-1)<0[/tex]
Take [tex]x=2\in(-2,\frac{9}{2})\ as\ -2<2<\frac{9}{2}[/tex]
[tex]f(2)=(2,-3)(2+2)(2+4)(2-1)(2\times2-9)\\\\=(-1)(4)(6)(1)(-5)\\\\=120\\\\f(2)>0[/tex]
Hence [tex]f(x)<0[/tex] for [tex]x=-1[/tex] and [tex]f(x)>0[/tex] for [tex]x=2[/tex]
So [tex]f[/tex] is sometimes positive and sometimes negative in the interval.
