Trigonometric Functions · Question 6

Let f: R R be a function defined by f(x)= cases x2 ( x2 ), & if x 0,\ 0, & if x=0. cases Then which of the following statements is TRUE?

1. f(x)=0 has infinitely many solutions in the interval [ 11010, ).
2. f(x)=0 has no solutions in the interval [ 1 , ).
3. The set of solutions of f(x)=0 in the interval (0, 11010 ) is finite.
4. f(x)=0 has more than 25 solutions in the interval ( 1 2, 1 ).

Answer

(4) f(x)=0 has more than 25 solutions in the interval ( 1 2, 1 ).

Solution

For x 0: x2=0 x2=n x= 1 n, n N.

(A) 1 n 10-10 n 1020: finitely many. False.

(B) 1 n 1 n 2 9.87: n=1, ,9 are solutions. False.

(C) n>1020: infinitely many solutions. False.

(D) 1 2< 1 n< 1 2<n< 4 97.4: n=10, ,97, i.e. 88 solutions >25. True.

Answer: (D)