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# Number of real solutions of the given equation where[] denotes gif and {} denotes fractional part function

Abhishek Singh
104 Points
one month ago
Solution.
$[8x^3]+[4x^2] +[2x] = \left \{ x \right \} -1 \\ \Rightarrow [8x^3]+[4x^2] +[2x] +1 = \left \{ x \right \}$

Now LHS is  an integer while RHS is a fractional part function ==> Solution exists if {x} = 0 or x is an integer
Now Assume x is an integer , we have to find solution of given equation
$[8x^3]+[4x^2] +[2x] +1 = 0 \Rightarrow 8x^3 +4x^2 +2x+1 = 0$                      as x is integer
Now 8x^3+ 4x^2+1 is monotonic increasing and f(-1) = -3 & f(0) =1 ==> the function has only one root b/w -1 and 0.
This is contradictory as our solution has to be integer

Thus the equation has no solution at all. [OPTION A]