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Question Number 160375 by HongKing last updated on 28/Nov/21

Commented by ghimisi last updated on 28/Nov/21

(1/(2b+1)) or (1/(4b+1))

$$\frac{\mathrm{1}}{\mathrm{2}{b}+\mathrm{1}}\:{or}\:\frac{\mathrm{1}}{\mathrm{4}{b}+\mathrm{1}} \\ $$$$ \\ $$

Commented by HongKing last updated on 28/Nov/21

Sorry my dear Ser, (1/(4b + 1))

$$\mathrm{Sorry}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Ser},\:\frac{\mathrm{1}}{\mathrm{4b}\:+\:\mathrm{1}} \\ $$

Answered by ghimisi last updated on 28/Nov/21

∀t∈R, (t^2 −t+1)(t^2 −2t+1)≥0⇒ t^4 +4t^2 +1≥3t^3 +3t t=(x^a /x^b )⇒x^(4a) +4x^(2a+2b) +x^(4b) ≥3x^(3a+b) +3x^(a+3b) ⇒∫_0 ^1 x^(4a) dx+4∫_0 ^1 x^(2a+2b) dx+∫_0 ^1 x^(4b) dx≥3∫_0 ^1 x^(3a+b) dx+3∫_0 ^1 x^(a+3b) dx ⇒(1/(4a+1))+(4/(2a+2b+1))+(1/(4b+1))≥(3/(3a+b+1))+(3/(a+3b+1))

$$\forall{t}\in{R},\:\left({t}^{\mathrm{2}} −{t}+\mathrm{1}\right)\left({t}^{\mathrm{2}} −\mathrm{2}{t}+\mathrm{1}\right)\geqslant\mathrm{0}\Rightarrow \\ $$$${t}^{\mathrm{4}} +\mathrm{4}{t}^{\mathrm{2}} +\mathrm{1}\geqslant\mathrm{3}{t}^{\mathrm{3}} +\mathrm{3}{t} \\ $$$${t}=\frac{{x}^{{a}} }{{x}^{{b}} }\Rightarrow{x}^{\mathrm{4}{a}} +\mathrm{4}{x}^{\mathrm{2}{a}+\mathrm{2}{b}} +{x}^{\mathrm{4}{b}} \geqslant\mathrm{3}{x}^{\mathrm{3}{a}+{b}} +\mathrm{3}{x}^{{a}+\mathrm{3}{b}} \\ $$$$\Rightarrow\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{4}{a}} {dx}+\mathrm{4}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{2}{a}+\mathrm{2}{b}} {dx}+\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{4}{b}} {dx}\geqslant\mathrm{3}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{3}{a}+{b}} {dx}+\mathrm{3}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{{a}+\mathrm{3}{b}} {dx} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{4}{a}+\mathrm{1}}+\frac{\mathrm{4}}{\mathrm{2}{a}+\mathrm{2}{b}+\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{4}{b}+\mathrm{1}}\geqslant\frac{\mathrm{3}}{\mathrm{3}{a}+{b}+\mathrm{1}}+\frac{\mathrm{3}}{{a}+\mathrm{3}{b}+\mathrm{1}} \\ $$$$ \\ $$

Commented by HongKing last updated on 28/Nov/21

Cool thank you so much my dear Ser

$$\mathrm{Cool}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Ser} \\ $$

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