Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 151905 by mathdanisur last updated on 24/Aug/21

Answered by mindispower last updated on 24/Aug/21

8x^x y^y z^z 2^(x+y+z) =2(2x)^x .2(2y)^y .(2z)^z_ 2(2t)^t ≥(t+1)^(t+1) ,t>0 ⇔2^(t+1) ≥t(1+(1/t))^(t+1) ⇔(t+1)ln(2)≥ln(t)+(t+1)ln(1+(1/t)) f(t)=(t+1)ln(2)−ln(t)−(t+1)ln(1+(1/t)) f′(t)=ln(2)−(1/t)−ln(1+(1/t))+(1/t)=ln(((2t)/(t+1))) f′(t)≥0,t∈[1,+∞[^� ,f′(t)<0,t∈]0,1[ ⇒f(t)≥f(1)=0 ⇒∀t>0,2.(2t)^(t+1) ≥(t+1)^(t+1) ⇒2.(2x)^(x+1) .2(2y)^(y+1) .2(2z)^(z+1) ≥(x+1)^(x+1) (y+1)^(y+1) (z+1)^(z+1) ⇔8x^x .y^y .z^z .2^(x+y+z) ≥(x+1)^(x+1) (y+1)^(y+1) (z+1)^(z+1)

8xxyyzz2x+y+z=2(2x)x.2(2y)y.(2z)z2(2t)t⩾(t+1)t+1,t>0⇔2t+1⩾t(1+1t)t+1⇔(t+1)ln(2)⩾ln(t)+(t+1)ln(1+1t)f(t)=(t+1)ln(2)−ln(t)−(t+1)ln(1+1t)f′(t)=ln(2)−1t−ln(1+1t)+1t=ln(2tt+1)Extra \left or missing \right⇒f(t)⩾f(1)=0⇒∀t>0,2.(2t)t+1⩾(t+1)t+1⇒2.(2x)x+1.2(2y)y+1.2(2z)z+1⩾(x+1)x+1(y+1)y+1(z+1)z+1⇔8xx.yy.zz.2x+y+z⩾(x+1)x+1(y+1)y+1(z+1)z+1

Commented by mathdanisur last updated on 24/Aug/21

Thank you Ser

ThankyouSer

Commented by mindispower last updated on 28/Aug/21

plrasur

plrasur

Terms of Service

Privacy Policy

Contact: info@tinkutara.com