Let A denote the set of all real numbers x such that x 3 - [ x ] 3 = ( x - [ x ] ) 3 , where [ x ] is the greatest integer less than or equal to x . Then

Question

Let A denote the set of all real numbers x such that x 3 - [ x ] 3 = ( x - [ x ] ) 3 , where [ x ] is the greatest integer less than or equal to x . Then, A is a discrete set of at least two points, A contains an interval, but is not an interval, A is an interval, but a proper subset of ( - ∞ , ∞ ), A = ( - ∞ , ∞ )

MARKS App · Accepted Answer

( x − [ x ] ) x 2 + [ x ] 2 + x [ x ] = ( x − [ x ] ) x 2 + [ x ] 2 − 2 x [ x ] ⇒ ( x − [ x ] ) ( 3 x [ x ] ) = 0 ⇒ x = 0 or [ x ] = 0 or x = [ x ] ⇒ x ∈ Z ∪ [ 0 , 1 )

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