设函数f(x)＝ex+x?a（a∈R，e为自然对数的底数），若曲线y=sinx上存在点（x0，y0）使得f（f（y0））=y0，
\u8bbe\u51fd\u6570f\uff08x\uff09=ex+x?a\uff08a\u2208R\uff0ce\u4e3a\u81ea\u7136\u5bf9\u6570\u7684\u5e95\u6570\uff09\uff0e\u82e5\u66f2\u7ebfy=sinx\u4e0a\u5b58\u5728\u70b9\uff08x0\uff0cy0\uff09\u4f7f\u5f97f\uff08f\uff08y0\uff09\uff09=y0\u7531\u9898\u610f\u53ef\u5f97 y0=sinx0\u2208[-1\uff0c1]\uff0cf\uff08y0\uff09=ey0+y0?a\uff0c\u2235\u66f2\u7ebfy=sinx\u4e0a\u5b58\u5728\u70b9\uff08x0\uff0cy0\uff09\u4f7f\u5f97f\uff08f\uff08y0\uff09\uff09=y0\uff0c\u2234\u5b58\u5728y0\u2208[0\uff0c1]\uff0c\u4f7ff\uff08y0\uff09=y0\u6210\u7acb\uff0c\u5373f\uff08x\uff09=x\u5728[0\uff0c1]\u4e0a\u6709\u89e3\uff0c\u5373 ex+x-x2=a \u5728[0\uff0c1]\u4e0a\u6709\u89e3\uff0e\u4ee4g\uff08x\uff09=ex+x-x2\uff0c\u5219a\u4e3ag\uff08x\uff09\u5728[0\uff0c1]\u4e0a\u7684\u503c\u57df\uff0e\u2235\u5f53x\u2208[0\uff0c1]\u65f6\uff0cg\u2032\uff08x\uff09=ex+1-2x\uff1e0\uff0c\u6545\u51fd\u6570g\uff08x\uff09\u5728[0\uff0c1]\u4e0a\u662f\u589e\u51fd\u6570\uff0c\u6545g\uff080\uff09\u2264g\uff08x\uff09\u2264g\uff081\uff09\uff0c\u53731\u2264a\u2264e\uff0c\u6545\u7b54\u6848\u4e3a\uff1a[1\uff0ce]\uff0e
\u89e3\uff1a\u7531f\uff08f\uff08b\uff09\uff09=b\uff0c\u53ef\u5f97f\uff08b\uff09=f-1\uff08b\uff09\u5176\u4e2df-1\uff08x\uff09\u662f\u51fd\u6570f\uff08x\uff09\u7684\u53cd\u51fd\u6570\u56e0\u6b64\u547d\u9898\u201c\u5b58\u5728b\u2208[0\uff0c1]\u4f7ff\uff08f\uff08b\uff09\uff09=b\u6210\u7acb\u201d\uff0c\u8f6c\u5316\u4e3a\u201c\u5b58\u5728b\u2208[0\uff0c1]\uff0c\u4f7ff\uff08b\uff09=f-1\uff08b\uff09\u201d\uff0c\u5373y=f\uff08x\uff09\u7684\u56fe\u8c61\u4e0e\u51fd\u6570y=f-1\uff08x\uff09\u7684\u56fe\u8c61\u6709\u4ea4\u70b9\uff0c\u4e14\u4ea4\u70b9\u7684\u6a2a\u5750\u6807b\u2208[0\uff0c1]\uff0c\u2235y=f\uff08x\uff09\u7684\u56fe\u8c61\u4e0ey=f-1\uff08x\uff09\u7684\u56fe\u8c61\u5173\u4e8e\u76f4\u7ebfy=x\u5bf9\u79f0\uff0c\u2234y=f\uff08x\uff09\u7684\u56fe\u8c61\u4e0e\u51fd\u6570y=f-1\uff08x\uff09\u7684\u56fe\u8c61\u7684\u4ea4\u70b9\u5fc5\u5b9a\u5728\u76f4\u7ebfy=x\u4e0a\uff0c\u7531\u6b64\u53ef\u5f97\uff0cy=f\uff08x\uff09\u7684\u56fe\u8c61\u4e0e\u76f4\u7ebfy=x\u6709\u4ea4\u70b9\uff0c\u4e14\u4ea4\u70b9\u6a2a\u5750\u6807b\u2208[0\uff0c1]\uff0c\u6839\u636eex+x?a\uff1dx\uff0c\u5316\u7b80\u6574\u7406\u5f97ex=x2-x+a\u8bb0F\uff08x\uff09=ex\uff0cG\uff08x\uff09=x2-x+a\uff0c\u5728\u540c\u4e00\u5750\u6807\u7cfb\u5185\u4f5c\u51fa\u5b83\u4eec\u7684\u56fe\u8c61\uff0c\u53ef\u5f97F(0)\u2264G(0)F(1)\u2265G(1)\uff0c\u5373e0\u226402?0+ae1\u226512?1+a\uff0c\u89e3\u4e4b\u5f971\u2264a\u2264e\u5373\u5b9e\u6570a\u7684\u53d6\u503c\u8303\u56f4\u4e3a[1\uff0ce]\u6545\u9009\uff1aA
曲线y=sinx上存在点（x0，y0）使得f（f（y0））=y0，则y0∈[-1，1]考查四个选项，B，D两个选项中参数值都可取0，C，D两个选项中参数都可取e+1，A，B，C，D四个选项参数都可取1，由此可先验证参数为0与e+1时是否符合题意，即可得出正确选项
当a=0时，f(x)＝
简单计算一下，答案如图所示 绛旓細鍥犫娍=(2a+1)^2-8a=(2a-1)^2鈮0 鍒檃x^2+(2a+1)x+2=0鑷冲皯鏈変竴涓牴 鍗宠〃鏄f(x)鍦≧涓婅嚦灏戞湁涓涓瀬鍊肩偣 鑻モ娍=0锛屽嵆a=1/2 姝ゆ椂1/2x^2+(2*1/2+1)x+2=0 鍗(x+2)^2=0 瑙ｅ緱x=-2 鏄剧劧褰搙<-2鍜寈>-2鏃讹紝閮芥湁(x+2)^2>0锛屽嵆f'(x)>0 琛ㄦ槑f(x)鍦≧涓婁负澧鍑芥暟 ... 绛旓細瑙ｏ細锛1锛塮鈥锛坸锛=ex锛坸-a锛塠x2+锛3-a+b锛墄+2b-ab-a]锛屼护g锛坸锛=x2+锛3-a+b锛墄+2b-ab-a锛屽垯鈻=锛3-a+b锛2-4锛2b-ab-a锛=锛坅+b-1锛2+8锛0锛屼簬鏄彲璁緓1锛寈2鏄痝锛坸锛=0鐨勪袱瀹炴牴锛屼笖x1锛渪2锛庘憼褰搙1=a鎴杧2=a鏃讹紝鍒檟=a涓嶆槸f锛坸锛鐨勬瀬鍊肩偣锛屾鏃朵笉鍚堥鎰... 绛旓細鍒欏垏绾跨殑鏂滅巼涓篹^t/t銆f'(x)=e^x锛屽垯鍒囩嚎鐨勬枩鐜囧張涓篺'(t)=e^t銆傛墍浠ワ紝e^t/t=e^t锛屽嵆t=1锛屽垏鐐逛负(1,e)銆佸垏绾挎枩鐜囦负e銆傚垏绾挎柟绋嬩负y=e(x-1)+e锛屽嵆y=ex銆傝嫢鍏充簬x鐨勬柟绋媏^x-ax=0鏈夊敮涓瑙ｏ紝鍒欐洸绾縴=e^x涓庣洿绾縴=ax鏈夊敮涓浜ょ偣銆傛墍浠ワ紝a鐨勫彇鍊艰寖鍥存槸(-鏃犵┓,0)U[e,+鏃犵┓)... 绛旓細鈭礷锛坸锛夆墺0瀵逛竴鍒噚鈭圧鎭掓垚绔嬶紝鈭-alna鈮0锛屸埓lna鈮0锛屸埓0锛渁鈮1锛屽嵆amax=1锛庯紙2锛璁緓1锛寈2鏄换鎰忕殑涓ゅ疄鏁帮紝涓攛1锛渪2锛屽垯g(x2)?g(x1)x2?x1锛瀖锛屾晠g锛坸2锛-mx2锛瀏锛坸1锛-mx1锛屸埓涓嶅Θ浠鍑芥暟F锛坸锛=g锛坸锛-mx锛屽垯F锛坸锛夊湪锛-鈭烇紝+鈭烇級涓婂崟璋冮掑锛屸埓F鈥诧紙x锛=g鈥诧紙... 绛旓細f(x)=e^x(x^2+ax+a+1)f'(x)=e^x(x^2+ax+a+1+2x+a)=e^x[x^2+(2+a)x+2a+1]=0 寰楋細x^2+(2+a)x+2a+1=0 delta=(2+a)^2-4(2a+1)=4+4a+a^2-8a-4=a(a-4)璁句笂杩版柟绋嬬殑鏍逛负x1,x2 褰揳>4鎴朼<0鏃讹紝x1,x2涓轰笉绛夊疄鏍癸紝鍑芥暟鏈変袱涓瀬鍊肩偣銆傚綋0=<a<=4... 绛旓細锛堚厾锛夌敱f锛坸锛=ex锛坸2+ax-a锛夛紝鍙緱f鈥诧紙x锛=ex[x2+锛坅+2锛墄]锛庘︼紙2鍒嗭級褰揳=1鏃讹紝f锛1锛=e锛宖鈥诧紙1锛=4e锛庘︼紙4鍒嗭級鎵浠ユ洸绾縴=f锛坸锛夊湪鐐癸紙1锛宖锛1锛夛級澶勭殑鍒囩嚎鏂圭▼涓簓-e=4e锛坸-1锛夛紝鍗硑=4ex-3e锛庘︼紙6鍒嗭級锛堚叀锛変护f鈥诧紙x锛=ex[x2+锛坅+2锛墄]=0锛岃В寰梮=... 绛旓細锛2锛夎瘉鏄庯細锛堚叞锛塯锛坸锛=f'锛坸锛=ex锛坸-a+1锛+锛坅-1锛夛紝g'锛坸锛=ex锛坸-a+2锛---锛5鍒嗭級褰揼'锛坸锛夛紲0鏃讹紝x锛渁-2锛涘綋g'锛坸锛夛紴0鏃讹紝x锛瀉-2 鍥犱负a锛2锛屾墍浠鍑芥暟g锛坸锛夊湪锛0锛宎-2锛変笂閫掑噺锛涘湪锛坅-2锛+鈭烇級涓婇掑---锛7鍒嗭級鍙堝洜涓篻锛0锛=0锛実锛坅锛=ea+a-1锛... 绛旓細鍥炵瓟锛氳璁篴鐨勫悇绉嶆儏鍐典笅 绛旓細閫堿锛岃鎯呭鍥炬墍绀 绛旓細涓銆佸崟璋冨尯闂村氨鏄f(x)姹傚 f 鈥锛坸锛=e^x-a,褰揻'(x)>0涓哄崟璋冨涔熷氨鏄痚^x-a>=0锛屽綋a<=0鏃跺繀鐒舵垚绔,涔熷氨鏄紙-鈭,+鈭烇級鍗曡皟澧 褰揳>0鏃讹紝鍒檟>=lna鍗曡皟澧炲尯闂翠负[lna,+鈭烇級锛屽悓鐞嗭細(-鈭,lna)涓哄崟璋冨噺鍖洪棿 浜屻乤=1,鍒欑敱绗竴姝ュ彲鐭ワ紙-鈭烇紝0锛夊崟璋冨噺锛岋紙0锛+鈭烇級鍗曡皟澧烇紙... 本站交流只代表网友个人观点，与本站立场无关 |
