The graph of the function f ( x )= x + 1 8 (2 x ), 0 x 1 is shown below. Define f₁(x)=f(x), f_ n+1 (x)=f (f_ n (x) ) , for n 1 Which of the following statements are true? I. There are infinitely many x [0,1] for which _ n f_ n (x)=0 II. The

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The graph of the function f ( x )= x + 1 8 (2 x ), 0 x 1 is shown below. Define f₁(x)=f(x), f_ n+1 (x)=f (f_ n (x) ) , for n 1 Which of the following statements are true? I. There are infinitely many x [0,1] for which _ n f_ n (x)=0 II. There are infinitely many x [0,1] for which _ n f _ n ( x )= 1 2 III. There are infinitely many x [0,1] for which _ n f_ n (x)=1 IV. There are infinitely many x [0,1] for which _ n f_

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Correct answer: B. II only

_ n f_ n (x)=f(f(f( times (x)) Now for x ₁ (0, 1 2 ) f ( x ₁ )> x ₁ as f ( x ) is concave - downward Thus f _ n 1 2 as n Similarly for x ₁ ( 1 2 , 1 ) f ( x ₁ ) < x ₁ as f ( x ) is concave upward Thus f _ n 1 2 as n

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