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a-b-c-1-a-b-c-1-Prove-that-1-a-2-1-1-b-2-1-1-c-2-1-27-10-

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Question Number 58964 by naka3546 last updated on 02/May/19
a + b + c = 1 a, b, c ≤ 1 Prove that (1/(a^2 + 1)) + (1/(b^2 + 1)) + (1/(c^2 + 1)) ≤ ((27)/(10))
a+b+c=1a,b,c⩽1Provethat1a2+1+1b2+1+1c2+1⩽2710
Answered by tanmay last updated on 02/May/19
((27)/(10))=((30−3)/(10))=(1+1+1)−((1/(10))+(1/(10))+(1/(10))) we have to prove (1/(a^2 +1))−1+(1/(b^2 +1))−1_ +(1/(c^2 +1))−1+(3/(10))<0 ((−a^2 )/(a^2 +1))+((−b^2 )/(b^2 +1))+((−c^2 )/(c^2 +1))+(3/(10)) (a−1)^2 >0 a^2 +1>2a (1/(a^2 +1))<(1/(2a)) (a^2 /(a^2 +1))<(a^2 /(2a)) −(a^2 /(a^2 +1))>−(a/2) −((a^2 /(a^2 +1))+(b^2 /(b^2 +1))+(c^2 /(c^2 +1)))>−((a/2)+(b/2)+(c/2)) −((a^2 /(a^2 +1))+(b^2 /(b^2 +1))+(c^2 /(c^2 +1)))+(3/(10))>((−1)/2)+(3/(10)) −((a^2 /(a^2 +1))+(b^2 /(b^2 +1))+(c^2 /(c^2 +1)))+(3/(10))>((−2)/(10)) right hand side is =((−1)/5)<0 so (1/(a^2 +1))+(1/(b^2 +1))+(1/(c^2 +1))≤((27)/(10)) others are requested for kind perusal please and check pls
2710=30−310=(1+1+1)−(110+110+110)wehavetoprove1a2+1−1+1b2+1−1+1c2+1−1+310<0−a2a2+1+−b2b2+1+−c2c2+1+310(a−1)2>0a2+1>2a1a2+1<12aa2a2+1<a22a−a2a2+1>−a2−(a2a2+1+b2b2+1+c2c2+1)>−(a2+b2+c2)−(a2a2+1+b2b2+1+c2c2+1)+310>−12+310−(a2a2+1+b2b2+1+c2c2+1)+310>−210righthandsideis=−15<0so1a2+1+1b2+1+1c2+1⩽2710othersarerequestedforkindperusalpleaseandcheckpls

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