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

$$\mathrm{For}\mathrm{x}\in (0;\mathrm{\infty })-\mathbb{Z}\mathrm{prove}\mathrm{that}:$$$$\frac{{\left\{\mathrm{x}\right\}}^3}{\left[\mathrm{x}\right]}+\frac{{\left[\mathrm{x}\right]}^3}{\left\{\mathrm{x}\right\}}⩾\frac{1}{8}({\mathrm{x}}^2+{\left[\mathrm{x}\right]}^2+{\left\{\mathrm{x}\right\}}^2)$$$$[\ast ]-\mathrm{GIF}\mathrm{and}\left\{\mathrm{x}\right\}=\mathrm{x}-\left[\mathrm{x}\right]$$

Commented by ghimisi last updated on 14/Oct/21

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

Commented by MathSh last updated on 14/Oct/21

$$\mathrm{Dear}\mathbf{S}\mathrm{er},\mathrm{try}\mathrm{for}\mathrm{x}>1$$

Commented by ghimisi last updated on 14/Oct/21

$$ok$$

Answered by ghimisi last updated on 14/Oct/21

$$x>1$$$$\left[x\right]=a⩾1$$$$\left\{x\right\}=b<1\Rightarrow x=a+b$$$$\iff 4(a^4+b^4)⩾ab(a^2+ab+b^2)\iff$$$$4(1+{\left(\frac{b}{a}\right)}^4)⩾{\left(\frac{b}{a}\right)}^3+{\left(\frac{b}{a}\right)}^2+\left(\frac{b}{a}\right)$$$$\frac{b}{a}=t<1$$$$\iff 4t^4+4⩾t^3+t^2+t$$$$t<1\Rightarrow t^3+t^2+t<3<4<4+4t^4$$

Commented by MathSh last updated on 14/Oct/21

$$\mathrm{Very}\mathrm{nice}\mathrm{dear}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}$$

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