Many Greatest Integers!

Algebra Level 4

Let f ( x ) = e x 2 − 4 + ⌊ e 2 x ⌋ + ⌊ e − 2 x ⌋ \large f(x) = e^{x^2-4} + \lfloor e^{2x} \rfloor + \lfloor e^{-2x} \rfloor .

What are the values of x x satisfying ⌊ f ( x ) ⌋ + ⌊ − f ( x ) ⌋ = 0 \lfloor f(x) \rfloor + \lfloor -f(x) \rfloor = 0 ?

− 2 , 2 -\sqrt { 2 } ,\sqrt { 2 } -1,1 -2,2 0

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Sahil Bansal by Sahil Bansal , 21, India

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1 solution

Deepak Kumar
Jan 8, 2016

With some observation& using the fact that floor(x)+floor(-x)=0 if x is an integer,we can easily eliminate other options and confirm x=2,x=-2 satisfy.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

f(x) is integer when portion of no truncate such that e x 2 − 4 = 0. e^{x^2 - 4} = 0.

Lu Chee Ket - 5 years, 4 months ago

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