构造对偶式证明不等式和恒等式小论文

()1x0xx111++xx23A=xx111++xxAB=xx111++xxA·B=1B231=A·B(23)AA0A23xx111++xx232a0b0ab=112+a12+b22A=12+a12+bAB=12+a12+bA2A2B2=4(ab)4=8A2212+a12+b223a1a2ana1a2an=12121aaa+3222aaa+12aaann+21M=2121aaa+3222aaa+12aaann+MN=2122aaa+3223aaa+nnnaaa+−12121aaan+MN=212221aaaa+−322322aaaa+−nnnnaaaa+−−−12211212aaaann+−=(a1a2)(a2a3)(a1−nan)(ana1)=(a1a2an)(a1a2an)=0M=Na2b221(a+b)2MN=212221aaaa++322322aaaa++nnnnaaaa++−−12211212aaaann++21(a1a2)21(a2a3)21(a1−nan)21(ana1)=a1a2an=12M1M212121aaa+3222aaa+12aaann+214n(1+1)(1+41)(1+231−n)313+nAn=(11)(141)(1231−n)=12·45·785343−−nn·2313−−nnAnBn=23·56·894333−−nn·133−nnCn=34·67·9103323−−nn·nn313+n1231−n=2313−−nn1131−n=133−nn1231−n=2313−−nn1n31=nn313+AnBn0AnCn0A3nAnBnCn=113+n=3n1(11)(141)(1231−n)313+n[]ABC22222sinsin2sinsincoscoscos2coscoscossinBCBCABCABCA+−=++=[][]2222cos,2sin,2sin,2sinbcbcAaaRAbRBcRC+−====()()22222222sinsin2sinsincoscoscos2coscoscos2sinsincos2coscoscoscossincossin2cossinsincoscoscos2cos22coscoscos2cos222222coscoscos2cos222BCBCABCABCBCAABCBBCCABCBCBCABCBCBCBCBC+−=++⇔−−=−+−⇔−+=+⇔−−=++−⇔=+[][][]222sinsin2sinsincossinxBCBCAA=+−−222coscos2coscoscossinyBCABCA=++−()()2222222222sinsin2sinsincossincoscos2coscoscossin112coscoscossinsin2sin22coscos2sin22cos2sin0xyBCBCAABCABCAABCBCAABCAAA+=+−−+++−=++−−=++−=−−=0xy−=0xy==[][]0,1,2...,ixin=12...1nxxx+++=22212122311...2nnxxxxxxxxx+++≥+++[][]()21121122xxxxxxλλ++≥+()22232232xxxxxxλλ++≥+()2112nnnnxxxxxxλλ++≥+()()222121223111...2...2...22nnnnxxxxxxxxxxxxxλλλλ++++++≥++−++=−21122222λλλ−=−−+14λ=1222λλ−=[][]222121223144...20nnxxxxxxxxx−+++×≤+++[]()222121223112231...nnnxxxfxxxxxxxxxxxxxxx=−++−+++−++++()22221212231...22nnxxxfxxxxxxxxx=+++−++++()0fx≥0Δ≤222121223144...20nnxxxxxxxxx−+++×≤+++22212122311...2nnxxxxxxxxx+++≥+++[][]21iiixxx++211iiixxx+++[]2221212231...nnxxxAxxxxxx=++++++22232112231...nxxxBxxxxxx=++++++22222223112122312311212.........1222nnnnxxxxxxABxxxxxxxxxxxxxxx++++=+++++++++≥+++=+++=222222231121223112231......0nnnxxxxxxABxxxxxxxxxxxx−−−−=+++=−+−++−=+++()21AABABAB=++−=+≥12A≥[][]911999999...0.00310121000000M=⋅⋅⋅[]99999910000000.60.00311mnmn−−⋅2122mnmn+−+1111log1log1,log122mnmn−+−−+()222221111112log1','011xxxxxxxxxx−−=⋅==−−−−1111log1log1log122mnmn−+−−+21111112mnmn−−−+499996911999999...10121000000999999911999997500002500006500004......10100000012999998500003500007500005500004500005M=⋅⋅⋅=⋅⋅⋅0.312399......234100⋅⋅12399......234100⋅⋅12399......234100⋅⋅911999999...10121000000⋅⋅⋅[]91011999998999999...1011129999991000000N=⋅⋅⋅⋅⋅1012999998911999999......1.......111399999910121000000=⋅⋅⋅⋅⋅⋅⋅1012999998......11113999999M=⋅⋅⋅⋅91000000N=0.003M2MN221012999998......111139999991012999998......111139999999119999991012999998.........11012100000011139999999101112999999,,.....,110111213100000011111,11101112MNMMM⇔⋅⋅⋅⋅⇔⋅⋅⋅⋅⇔⋅⋅⋅⋅⋅⋅⋅⇐⇐−−−11,......,110131000000−−−[]9119999991012999998...,......1101210000001113999999⋅⋅⋅⋅⋅⋅⋅[]12,xx230xx+−=32124190xx−+=[][]12,xx230xx+−=22112230,30xxxx+−=+−=2211223,3xxxx=−=−()()()()32121122112112121241934319312419334744444xxxxxxxxxxxxxxx−+=−−−+=−−++=−−++=++=++121xx+=−3212419440xx−+=−+=[][]12,xx[]3212419Mxx=−+3221419Nxx=−+()()()()()()332212122212121212124383423819416380MNxxxxxxxxxxxxxx+=+−++=++−−+−+=−+−++=()()()()()()()()3322121222121212122121212121244441340MNxxxxxxxxxxxxxxxxxxxxxx−=−+−=−++++=−+−++=−+−=()102MMNMN=++−=[][][]xx()66510581+=[]()665+[]()()6665,65MN=+=−1MN=01N()()()()()()()()()()()()()()()3333222222222656565656565656536565656565656565262524324119323021119787201119575611058111197872210582111978724MNMNMNMN−=++−+−−=++−++−−+−×+−+++−−+−=×−−=⇒−=−+=⇒==+=1058110582M[]10581M=[]MN−M193230MN=+M19323010581.9998
N10582[]M[]121coscos2...cos2coscos22nnnnnnxnxMCxCxCnx=++++=[]M()111221cos...cos1...1nnnnnnnnnMCxCnxCtCtCtt=+++=++++=+cos,kkxt()cossincossinkkxikxxix+=+[]121coscos2...cosnnnnMCxCxCnx=++++12sinsin2...sinnnnnNCxCxCnx=+++()()()()()()()122121cossincos2sin2...cossin1cossincossin...cossin1cossinnnnnnnnnnnMNiCxixCxixCnxinxCxixCxixCxixxix+=+++++++=+++++++=++21cossin2cos2sincos2coscossin222222xxxxxxxixii++=+=+2coscossin2coscos22222nnnnxnxnxxnxMNiiM+=+⇒=[][]223cos10cos50sin40sin804+−=oooo[][]()()22222222cos10cos50sin40sin801sin80sin402sin40sin802sin80sin402cos1804080sin40sin80sin18040803234+−=+−××=+−×−−×=−−==oooooooooooooooooo[][][]2222cos10cos50sin40sin80sin10sin50cos40cos80MN=+−=+−oooooooo()()11sin40sin80cos40cos802cos80402cos40MN+=+−+=−−=−ooooooo()()()2222cos10sin10cos50sin50cos40cos80sin40sin80cos20cos100cos40801cos20cos8028020208012sinsin2221sin502MN−=−+−+−=+++=−−−+=−=−ooooooooooooooooooo()11132cos40sin502224MMNMN=++−=−+−=oo[][]227cos20cos803cos20sin804++=oooo[][]2222sin20sin803sin20cos80cos20cos803cos20sin80MN=++=++oooooooo()113cos20sin80sin20cos80113sin10023sin80MN+=+++=++=+oooooo()()()()2222cos20sin20cos80sin803sin80cos20sin20cos80cos40cos1