Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

Question Number 50890 by peter frank last updated on 21/Dec/18

Determine whether the following is true for all value of x 0≤(((x+1)^2 )/(x^2 +x+1))≤(4/3)

$${Determine}\:{whether} \\ $$$${the}\:{following}\:\:{is}\:{true}\:{for}\:{all} \\ $$$${value}\:{of}\:{x} \\ $$$$\mathrm{0}\leqslant\frac{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\leqslant\frac{\mathrm{4}}{\mathrm{3}} \\ $$

Answered by MJS last updated on 21/Dec/18

∀x∈R: (x+1)^2 ≥0∧(x^2 +x+1)≥0 ⇒ 0≤(((x+1)^2 )/(x^2 +x+1)) 3(x+1)^2 ≤4(x^2 +x+1) 3x^2 +6x+3≤4x^2 +4x+4 0≤x^2 −2x+1 0≤(x−1)^2 ∀x∈R ⇒ ⇒ 0≤(((x+1)^2 )/(x^2 +x+1))≤(4/3) ∀x∈R

$$\forall{x}\in\mathbb{R}:\:\left({x}+\mathrm{1}\right)^{\mathrm{2}} \geqslant\mathrm{0}\wedge\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\geqslant\mathrm{0}\:\Rightarrow\:\mathrm{0}\leqslant\frac{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{x}+\mathrm{1}} \\ $$$$\mathrm{3}\left({x}+\mathrm{1}\right)^{\mathrm{2}} \leqslant\mathrm{4}\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right) \\ $$$$\mathrm{3}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{3}\leqslant\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{4} \\ $$$$\mathrm{0}\leqslant{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1} \\ $$$$\mathrm{0}\leqslant\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\forall{x}\in\mathbb{R}\:\Rightarrow \\ $$$$\Rightarrow\:\mathrm{0}\leqslant\frac{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\leqslant\frac{\mathrm{4}}{\mathrm{3}}\:\forall{x}\in\mathbb{R} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com