For-x-0-Z-prove-that-x-3-x-x-3-x-1-8-x-2-x-2-x-2-GIF-and-x-x-x-

Question Number 156642 by MathSh last updated on 13/Oct/21

For x \in (0; \infty) - Z prove that :

\frac{{x}^{3}}{[x]} + \frac{{[x]}^{3}}{{x}} ⩾ \frac{1}{8} (x^{2} + {[x]}^{2} + {x}^{2})

[*] - GIF and {x} = x - [x]

Commented by ghimisi last updated on 14/Oct/21

x \in (0, 1) \Rightarrow [x] = 0 ? ? ? ? ? ? ? ? ?

Commented by MathSh last updated on 14/Oct/21

Dear S er, try for x > 1

Commented by ghimisi last updated on 14/Oct/21

o k

Answered by ghimisi last updated on 14/Oct/21

x > 1

[x] = a ⩾ 1

{x} = b < 1 \Rightarrow x = a + b

\Leftrightarrow 4 (a^{4} + b^{4}) ⩾ a b (a^{2} + a b + b^{2}) \Leftrightarrow

4 (1 + {(\frac{b}{a})}^{4}) ⩾ {(\frac{b}{a})}^{3} + {(\frac{b}{a})}^{2} + (\frac{b}{a})

\frac{b}{a} = t < 1

\Leftrightarrow 4 t^{4} + 4 ⩾ t^{3} + t^{2} + t

t < 1 \Rightarrow t^{3} + t^{2} + t < 3 < 4 < 4 + 4 t^{4}

Commented by MathSh last updated on 14/Oct/21

Very nice dear S er, thank you

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