tmua.me Extreme Set C Paper 2 · Question 13

For $0<x<1$ and each positive integer $n$ , define

F_n(x)=nx^n(1-x).

Consider the following statements.

For every real $x$ with $0<x<1$ , there exists a positive integer $N$ such that $F_n(x)<1/4$ for every integer $n\geq N$ .
There exists a positive integer $N$ such that $F_n(x)<1/4$ for every real $x$ with $0<x<1$ and every integer $n\geq N$ .
For every positive integer $n$ , there exists a real $x$ with $0<x<1$ such that $F_n(x)\geq1/4$ .

Which of these statements are true?

AI only

BIII only

CI and II only

DII and III only

EI and III only

FI, II and III

examday