The correct statements is (are) f'(1)<0, f'(x)≠0 for any x∈(1,3), f(2)<0, f'(x)≠0 for some x∈(1,3)

f(x)=xF(x)
Differentiating both side w.r.t x
f'(x)=F(x)+xF'(x)
∴f'(1)=F(1)+F'(1)=0+F'(1)<0
⋲F'(x)<0 ∀x∈(1,3)
f(2)=2F(2)<2F(1)<0, Since F(x) is decreasing function in the given interval
also f'(x)<0 ∀x∈(1,3) ⇒f(x)≠0 for any x∈(1,3)