Let-a-b-0-and-a-1-b-1-a-b-2-Prove-that-a-b-a-1-3-b-1-3-a-1-2-b-1-2-1-2-a-1-3-b-1-3- Tinku Tara June 4, 2023 Algebra 0 Comments FacebookTweetPin Question Number 145314 by loveineq last updated on 04/Jul/21 Leta,b⩾0and(a+1)(b+1)=(a+b)2.Provethat(a+b)(a+1)3+(b+1)3⩽(a+1)2+(b+1)2⩽12[(a+1)3+(b+1)3] Commented by justtry last updated on 05/Jul/21 (a+b)(b+1)=(a+b)2⇔ab+a+b+1=a2+2b+b2⇔1+(a+b)=a2+2b+b2b=0⇒(1+a)=a2a=0⇒(1+b)=b2&(1+a)3⩾(1+a)2=(a2)2(1+b)3⩾(1+b)2=(b2)212[(1+a)3+(1+b)3]⩾12[(a2)2+(b2)2]⩾(a2+b2)24⇔12[(1+a)3+(1+b)3]⩾((a2)2+(b2)2)⩾(a2+b2)22⇔12[(1+a)3+(1+b)3]⩾12(a4+b4+2a2b2)=12[(1+a)2+(1+b)2+2(a+b)2]note12[(1+a)2+(1+b)2+2(a+b)2]⩾(1+a)2+(1+b)2→a=b=1ora≠b≠1,a,b⩾0so12[(1+a)3+(1+b)3]⩾(1+a)2+(1+b))2⇔(1+a)2+(1+b)2⩽12[(1+a)3+(1+b)3]……… Commented by loveineq last updated on 05/Jul/21 why1+a+b=a2+2b+b2? Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-145309Next Next post: Let-f-0-1-R-be-a-differentiable-function-such-that-f-f-x-x-for-all-x-0-1-and-f-0-1-If-n-is-a-positive-integer-evaluate-the-following-integral-0-1-x-f-x-2n Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.