Question

help me prove it by using harmonic mean-geometric mean-arithmetic mean-quadratic mean. please give complete an clear steps to understand. Thank you

Prove the HM-GM-AM-QM inequalities! . Let X, X₂, ..., xn be positive real numbers Prove that: i x, X, X . min x X₂ ..., xnyt n n <max {x,,x2,...,xng

Answer 1

We have to prove that, min far, X2, 3 days latent sx xn. 22x2²to antan C-24 +22totan Caxtor totes max hau, 2, ..) n a Let xà be the minimum number of Exi! (=1, 2, 2, 3 jnow xáca I 3 I Fitxi Zã XY Vai I dan test cular t o PjXhos IX X obowegwe von go three montempo ? uties Anabel by 37 tonton adding , (A) to be --utoa equinity holds when nex2= m . wow coe seloos chave to show, x1 * x2 + 3 + xin nga X2 ...n x eno veusso staliupa & ab Here, 2,22 2 1 84 +22) ² - ( 91 922) 2 a (24 + H2) 2 since (2, -8 29 30 2 equinity holds when 21222 2 emilankyo Ag $ 108,) egninity occurs when agad.

Therefore, T. 27 X2 s -2 42 4 4+420 2 21X2X384 3 (x4+x2)? ( 23+24)? now for 24+ 22 and 23+24 forum incanlity 2*X3 Xatala +22+ X2 +48)? so, from ineanility 2 wę can. r4 X2 X3 X4 Ś 12 egninity holds when a = p X3 zda, de similarly go xs X7 Xe s fast26 & 27 tag) x 4 xq xp e la CI 4 Proceeding with similar anguments we have X1 X2 X3. - Ros (x+d2trada S8 the eawility occurs when x=222 nadg Continuing this process when neem where m is positive ibrtager y we geto per dig ensamo solo company stories ware equshity occurs when X 182.- Un Gitaeteretan o O equility occurs when XFX2anomaan legt das

Let us consider mon not at po power of 2 Then I non sit amals ng 2l2m, m 22 m . MEIN & to L So, I PERIN site n&p=2m. snow we stake on AP Integers sutil et ( ) ( ) 2 - m 2 , 7 , 7 - et times where x 2 344 X 2f a tan since nap=2m, we can use the previous part at the inequility and we can write , A s an a fetta26 tantoa egronity occurs when a a nsa. X,X2 . . Xnxb a l'art de tot ant ponto = nod. o equility occurs aren a=X2E NAP 20482... an x smxtpal . nap 2 9 2,22--Ky a NAP Elx (mtp)). mtp 44X2--Xn op & antip 33.22. - 29. Sexm & Z. (X4 X2 ... Xon) in g x4+x2 +*-*tan z 18182--- XN) YA...B eauility occurs when x=X2= naan so we can say A.M I G.M.

Inow we have to peroren 126, 2 91 291 (84X2 + x n ) I It the front an L olu S Here since a à length The A.m. At t grabenor are positive. are also positive feet h em to 222 Tanong Warnere for n est and Gom. at o nton is INNR n toin' X from previous part , A.M. I Grim.. 2,792mmän > I dio)". eaninty holds when 22. Sex, Sedan - entre I torer 9 (2422 --Xn) yn > ^ cao Goxi S tipete

2 or snow Cauchy o Schwarzenequinty If xr, dan an and J, 42, er ty are alla real numbers then @(0,+x2 + .--tam?) 147+82 + ... + 8 m2) 21 MY+X292 tartantno equility occurs when ai = 0,1 21,2,..., or biso , i=1,2,. n or a sbico, is 1, 2, 3 or a; ak bi, 121, 2) ...n seinere K is non 200 Уступить , , We use this cauchy - Schwarz, frenuinity to prove, Xitxatient anĮ x 24 x 2 foretag² Let us choase, the real numbers also xi, Xajares an and Ilonnel in times). АГФК oppi rg CM Wy - Гun 00М2 n o (м и (© X rinn (?x2 int xonon(1 321 puti2) S (xi txetinet and 2 3) au+Xq tort sem se zno (2,34 xz2torot xp?j V2 ... x2 & X2Ztractant ---. (D) 8 xit dat natan an ? xeaminky holds when x=42=0.axn.

snow we have to show roroo orood gas Ja? +2²4. Ant į max hai, K2, rydns. V m Let x; a mäx qx, 2..ang. 2 mm and we areas X; ? X2 *;? Zazd ro Car & ai n gx. 2 x 2 y = 10 X; = X; W;? Za; by adding by adding all at this na; 2 2 x 2 + 2 2 2 2 + x 2 x 9 Japonesa... A Xin? xzaman {1,495,-Moniz Info Now from the emota incaminties, (A), B), C), 002,,(E) we can say, min h 21,812 -n 35 vastatud ka "/W, 22-an minta tolon e 242 ... si max hon yn 2002, an?. 19 tat..tan hentan ..7010 s n