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

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

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Grade:12

1 Answers

Abhishek Singh
105 Points
2 years 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]
 
 
 

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