若函数f(x)为偶函数，且y=f(x+1)为奇函数，f(2)=-1，则f(1)+f(2)+..... 若函数f（1+x）为偶函数，f（x-2）为偶函数，则y=f（...
\u82e5\u51fd\u6570f\uff08x\uff09\u7684\u5b9a\u4e49\u57df\u4e3aR\uff0c\u4e14\u6ee1\u8db3y=f\uff08x+1\uff09\u4e3a\u5947\u51fd\u6570\uff0cy=f\uff08x-1\uff09\u4e3a\u5076\u51fd\u6570\uff0c\u5219\u4e0b\u5217\u8bf4\u6cd5\u4e2d\u4e00\u5b9a\u6b63\u786e\u7684\u6709___\uff081\uff09\u2235y=f\uff08x-1\uff09\u4e3a\u5076\u51fd\u6570\uff0c\u5373\u5bf9\u79f0\u8f74\u4e3ax=0\uff0c\u5c06y=f\uff08x-1\uff09\u7684\u56fe\u8c61\u5411\u5de6\u5e73\u79fb\u4e00\u4e2a\u5355\u4f4d\u5373\u5f97y=f\uff08x\uff09\u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebfx=-1\u5bf9\u79f0\uff0c\u2234\u6545\uff081\uff09\u6b63\u786e\uff1b\uff082\uff09y=f\uff08x+1\uff09\u4e3a\u5947\u51fd\u6570\uff0c\u5373\u5bf9\u79f0\u4e2d\u5fc3\u4e3a\uff080\uff0c0\uff09\uff0c\u5c06y=f\uff08x+1\uff09\u7684\u56fe\u8c61\u5411\u53f3\u5e73\u79fb\u4e00\u4e2a\u5355\u4f4d\u5373\u5f97y=f\uff08x\uff09\u7684\u56fe\u8c61\u5173\u4e8e\u70b9\uff081\uff0c0\uff09\u5bf9\u79f0\uff0c\u2234f\uff08-x\uff09=-f\uff08x+2\uff09\u2460\u2235y=f\uff08x\uff09\u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebfx=-1\u5bf9\u79f0\uff0c\u2234f\uff08-x\uff09=f\uff08x-2\uff09\u2461\u7531\u2460\u2461\u53ef\u5f97\uff0cf\uff08x+2\uff09=-f\uff08x-2\uff09\uff0c\u5c06x\u4ee3\u6362\u4e3ax+2\uff0c\u5f97f\uff08x+4\uff09=-f\uff08x\uff09\uff0c\u518d\u5c06x\u4ee3\u6362\u4e3ax+4\uff0c\u5f97f\uff08x+8\uff09=-f\uff08x+4\uff09=f\uff08x\uff09\uff0c\u2234f\uff08x\uff09\u7684\u5468\u671f\u4e3a8\uff0c\u6545\uff082\uff09\u9519\u8bef\uff1b\uff083\uff09\u6839\u636e\u51fd\u6570\u7684\u5468\u671f\u4e3a8\uff0c\u2234f\uff082013\uff09=f\uff085\uff09\uff0c\u2235f\uff08x+4\uff09=-f\uff08x\uff09\uff0c\u2234f\uff085\uff09=-f\uff081\uff09\u2235f\uff08-x\uff09=-f\uff08x+2\uff09\uff0c\u6240\u4ee5f\uff081\uff09=-f\uff081\uff09\u5373f\uff081\uff09=0\uff0c\u2234f\uff082013\uff09=0\uff0c\u6545\uff083\uff09\u6b63\u786e\uff1b\uff084\uff09\u7531\uff083\uff09\u53ef\u77e5f\uff081\uff09=f\uff08-3\uff09=f\uff085\uff09=0\uff0c\u82e5\u5f53f\uff080\uff09=0\u65f6\uff0c\u5219f\uff082\uff09=0\uff0cf\uff08-2\uff09=0\u6545f\uff08x\uff09\u5728[-2\uff0c2]\u4e0a\u4e0d\u6b62\u4e00\u4e2a\u96f6\u70b9\uff0e\u6545\uff084\uff09\u4e0d\u4e00\u5b9a\u6b63\u786e\u6545\u7b54\u6848\u4e3a\uff1a\uff081\uff09\uff083\uff09\uff0e
\u7279\u5f811\uff0c\u5bf9\u79f0\u8f74\uff1a
\u5bf9\u79f0\u51fd\u6570\u4f53\u73b0\u4e3a\uff0cf(a+x)=f(a-x)\u65f6\uff0cf(x)\u5173\u4e8ex=a\u8f74\u5bf9\u79f0\uff0c
f(1+x)\u4e3a\u5076\u51fd\u6570\uff0cf(1+x)=f(1-x)\uff0c\u6240\u4ee5f(x)\u5173\u4e8ex=1\u8f74\u5bf9\u79f0\uff0c
f(x-2)\u4e3a\u5076\u51fd\u6570\uff0cf(x-2)=f(-2+x)=f(-2-x)\uff0c\u6240\u4ee5f(x)\u5173\u4e8ex=-2\u8f74\u5bf9\u79f0\uff0c
\u7279\u5f812\uff0c\u5468\u671f\uff1a
\u5982\u679cf(x)=f(x+T)\uff0c\u5219\u51fd\u6570f(x)\u662f\u5468\u671f\u4e3aT\u7684\u5468\u671f\u51fd\u6570\uff0c
f(1+x)=f(x+1)=f((x+3)-2)=f(-2-(x+3))=f(-5-x)=f(1+(-6-x))=f(1-(-6-x))=f(7+x)=f(x+7)\uff0c
\u51fd\u6570f(x)\u662f\u5468\u671f\u51fd\u6570\uff0c\u5468\u671fT=7-1=6\uff0c
f(x-1)=-f(-x-1) x用x+1代 f(x)=-f(-x-2)=f(-x) , -x用x代 -f(x-2)=f(x)
再写一个 -f(x-4)=f(x-2) 上两式一合并 得f(x)=f(x-4),周期为4
f(1)=f(-1)=0,f(2)=-1,f(3)=f(-1)=0,f(4)=f(0)=-f(-2)=-f(2)=1
故有f(1)+f(2)+f(3)+f(4)=0-1+0+1=0
2014/4=503...2
故有f(1)+f(2)+...+f(2014)=f(1)+f(2)=0-1=-1
f(1)+f(2)+.....f(1)+f(2)+......+f(2014)是一直加到f(2014)？
绛旓細f(x)涓哄伓鍑芥暟锛鍙煡f(x)=f(x)=f(-x) f(x)褰撶劧绛変簬f(|x|)鍙互鐞嗚В涓哄伓鍑芥暟鐨勫浘鍍忓叧浜巠杞村绉帮紝鏃犺x涓烘鎴栦负璐燂紝鍑芥暟鍊糵锛坸锛夊间笉鍙樸傚嵆f(x)=f(|x|)
绛旓細浠x=y=0, 寰 f(0)=2f(0)+0, f(0)=0 浠=-x, 寰 f(0)=f(x)+f(-x)-x²f(-x)=f(x)鎵浠ワ細 f(x)=x²/2 楠岃瘉锛歠(x+y)=x²/2+y²/2+xy=(x+y)²/2
绛旓細瑙ｆ瀽锛氣埖y=f(x)鏄畾涔夊湪R涓婄殑鍋跺嚱鏁,涓攆(x)=f(x+3)鈭磃(x)鏄渶灏忔鍛ㄦ湡涓3鐨勫懆鏈熷嚱鏁帮紝f(-x)=f(x)鈭鑻ュ嚱鏁皔=f(x)鍥惧儚鏃㈠叧浜庣偣A(a,c)鎴愪腑蹇冨绉板張鍏充簬鐩寸嚎x=b鎴愯酱瀵圭О锛坅鈮燽锛夛紝鍒檡=f(x)鏄懆鏈熷嚱鏁帮紝涓4|a-b|鏄叾涓涓懆鏈熴F(x) 鍏充簬鐩寸嚎x=0鎴愯酱瀵圭О锛屽叧浜庣偣A(a,0)...
绛旓細澶栧鍐呭涓哄,澶栧鍐呭伓涓哄伓,澶栧伓鍐呭涓哄伓,澶栧伓鍐呭伓涓哄伓.F=f(g(X)),鑻(X)涓哄伓鍑芥暟锛褰撲换鎰忓彇鍏充簬X瀵圭О鐨勪袱鐐筙1,-X1鏃讹紝鏈塯(X1)=g(-X1)锛屾墍浠(g(X1))=f(g(-X1))銆侳涓哄伓鍑芥暟锛屽洜姝ゅ唴鍋跺垯鍋躲 F=f(g(X)),鑻(X)涓哄鍑芥暟锛屽綋浠绘剰鍙栧叧浜嶺瀵圭О鐨勪袱鐐筙1,-x1鏃讹紝鏈-g(X1...
绛旓細鍋跺嚱鏁 鍒檉(-x)=f(x)寰梐=c=0 f(x)=x^4+b*x^2+d 鍦▁=1鍑哄彇鏋佸-1 鍗砯(1)=-1鍜宖'(1)=0 f'=4x^3+2bx 鈭 1+b+d=-1 4+2b=0 寰梑=-2 d=0 y=x^4-2x^2
绛旓細鍋跺嚱鏁鍒欏绉拌酱鏄痻=0 y=ax²+(1-a²)x-a 鍒-(1-a²)/2a=0 a=卤1 >0閫掑鍒欏紑鍙ｅ悜涓 鎵浠>0 鎵浠=1
绛旓細鍋跺嚱鏁帮紝鍏充簬y杞村绉帮紝x<0鏃讹紝f(x)閫掑锛岀敾鍑鸿崏鍥剧敱瀵圭О鎬у彲鐭ワ細x>0鏃讹紝f(x)閫掑噺锛涘浘鍍忕被浼间簬涓涓紑鍙ｅ悜涓嬬殑鎶涚墿绾匡紝绂诲绉拌酱瓒婅繙锛屽嚱鏁板艰秺灏忥紝绂诲绉拌酱瓒婅繎锛屽嚱鏁板艰秺澶с傚埌瀵圭О杞寸殑璺濈鐢ㄧ粷瀵瑰兼潵琛￠噺 瑕佷娇寰梖(x+1)>f(1-2x)锛屽垯x+1鍒板绉拌酱鐨勮窛绂绘瘮1-2x鍒板绉拌酱鐨勮窛绂昏繎锛涘嵆|x+...
绛旓細杩愮畻娉曞垯 (1) 涓や釜鍋跺嚱鏁扮浉鍔犳墍寰楃殑鍜涓哄伓鍑芥暟銆(2) 涓や釜濂囧嚱鏁扮浉鍔犳墍寰楃殑鍜屼负濂囧嚱鏁般(3) 涓涓伓鍑芥暟涓庝竴涓鍑芥暟鐩稿姞鎵寰楃殑鍜屼负闈炲鍑芥暟涓庨潪鍋跺嚱鏁般(4) 涓や釜鍋跺嚱鏁扮浉涔樻墍寰楃殑绉负鍋跺嚱鏁般(5) 涓や釜濂囧嚱鏁扮浉涔樻墍寰楃殑绉负鍋跺嚱鏁般(6) 涓涓伓鍑芥暟涓庝竴涓鍑芥暟鐩镐箻鎵寰楃殑绉负濂囧嚱鏁般
绛旓細涓瀹氭病鏈 鍋跺嚱鏁瀵逛簬涓涓獃瀵瑰簲2涓笉鍚岀殑x锛岄偅涔堜粬鐨勫弽鍑芥暟瀵逛簬涓涓獂灏辨湁2涓笉鍚岀殑y锛堝弽鍑芥暟灏辨槸浜ゆ崲x,y鍢涳級锛岃繖杩濆弽浜嗗嚱鏁扮殑瀹氫箟锛屾墍浠ユ病鏈 鏈夛紝姣斿y=x^2鍦紙0锛岋紞鏃犵┓澶э級灏辨湁鍙嶅嚱鏁 杩欎釜鏄笉瀵圭殑锛屽伓鍑芥暟瀹氫箟鏄浜庝竴涓f(x)鏈塮(x)=f(-x) f(-x)閮芥病鏈夊畾涔変簡杩樻槸鍋跺嚱鏁板悧锛
绛旓細闄ら潪褰鍑芥暟y=f(x)锛 瀹氫箟鍩熸槸{0} 涓 f(x)=C 锛堝叾涓瑿鏄父鏁帮級鏃讹紝璇鍑芥暟f(x)鏄鍋跺嚱鏁颁笖鏈夊弽鍑芥暟锛鍏跺弽鍑芥暟鐨勫畾涔夊煙鏄瘂C}锛屽煎煙涓簕0}銆傝屽浜庡鍑芥暟鑰岃█锛屽鍑芥暟鏈韩鍏充簬鍘熺偣瀵圭О锛屽叾鍙嶅嚱鏁板張鍏充簬y=x瀵圭О锛屾墍浠ヨ嫢涓涓鍑芥暟瀛樺湪鍙嶅嚱鏁帮紝鍒欏畠鐨勫弽鍑芥暟涔熸槸濂囧嚱鏁般
