设fx在〔0，1〕上连续，在（0，1）内可导，且f（1）=0,求证:存在a属于（0，1），fa的导 设函数f(x)在区间［0,1］上连续，在区间（0，1）内可导...
\u8bbef(x)\u5728\uff3b0,1\uff3d\u4e0a\u8fde\u7eed\uff0c\u5728\uff3b 0,1\uff3d\u5185\u53ef\u5bfc\uff0c\u4e14f\uff080\uff09=1\uff0cf\uff081\uff09=0\uff0c\u8bc1\u660e:\u5728\uff080,1\u8bbeF(x)=xf(x)
\u56e0\u4e3a f(x)\u5728\u533a\u95f4[0,1]\u4e0a\u8fde\u7eed\uff0c\u5728\u533a\u95f4(0,1)\u5185\u53ef\u5bfc
\u5f97F(x)\u5728\u5728\u533a\u95f4[0,1]\u4e0a\u8fde\u7eed\uff0c\u5728\u533a\u95f4(0,1)\u5185\u53ef\u5bfc
\u4e14F'(x)=f(x)+xf'(x)
\u53c8f(1)=0 ,\u5f97F(0)=F(1)=0
\u6839\u636e\u7f57\u5c14\u5b9a\u7406\u5f97
\u5b58\u5728a\u2208(0,1),\u4f7fF'(a)=(a)+af'(a)=0
\u6240\u4ee5\u5b58\u5728a\u2208(0,1),\u4f7ff(a)+af'(a)=0
\u5e0c\u671b\u80fd\u5e2e\u5230\u4f60\uff01
绛旓細濡傛灉f(0)=0锛屽垯鍙杄=0銆傚鏋渇(1)=1锛屽彇e=1銆傚鏋渇(0)鈮0锛宖(1)鈮1锛屼护F(x)=f(x)-x锛屽垯F(x)鍦╗0,1]涓婅繛缁紝F(0)=f(0)-0锛0锛孎(1)=f(1)-1锛0锛岀敱闆剁偣瀹氱悊锛屽瓨鍦╡鈭(0,1)锛屼娇寰桭(e)=0锛屽嵆f(e)=e銆傜患涓婏紝瀛樺湪e鈭圼0,1]锛屼娇寰梖(e)=e銆
绛旓細涓嶄竴瀹氥備緥濡俁涓婂懆鏈烼=2鐨勫嚱鏁癴(x)锛屽綋-1鈮锛1鏃秄锛坸锛=x锛屼綔鍥惧彲鐭 |f(x)|杩炵画锛鑰宖(x)鍦ㄦ墍鏈夊鏁扮偣涓嶈繛缁
绛旓細搴旂敤瀹氱Н鍒嗕腑鍊煎畾鐞嗭細瀛樺湪尉1鈭锛0锛1锛変娇寰 鈭(0鈫1)f(x)dx=f(尉dao1)(1-0)=f(尉1)鎵浠ワ紝f(尉1)=f(2) 鍐嶆搴旂敤缃楀皵瀹氱悊锛屽瓨鍦ㄎ锯垐锛埼1锛2锛夈愬綋鐒段锯垐锛0锛2锛夈 浣垮緱锛歠'(尉)=0 涓鑸畾鐞 瀹氱悊1锛氳f(x)鍦ㄥ尯闂碵a,b]涓婅繛缁紝鍒檉(x)鍦╗a,b]涓婂彲绉傚畾鐞2锛氳f(x...
绛旓細璁緂(x)=f(x)-x²+x/2 g(0)=f(0)-0+0=0 g(1/2)=f(1/2)-1/4+1/4=0 g(1)=f(1)-1+1/2=0.5-1+1/2=0 鍥犳g(x)鍦╗0锛1]鍐呮湁涓変釜闆剁偣锛屼笖g(x)鏄剧劧鏄簩闃跺彲瀵肩殑 鐢辩綏灏斿畾鐞嗭細瀛樺湪畏1鈭(0锛1/2)锛屛2鈭(1/2锛1)浣匡細g'(畏1)=0锛実'(畏2)=0 ...
绛旓細璁捐繃A ,B 鐨勭洿绾垮嚱鏁颁负y=g(x)鍒檉(0)=g(0) f(c)=g(c) f(1)=g(1)鐢辨媺鏍兼湕鏃ヤ腑鍊煎畾鐞嗗緱锛歔f(c)-f(0)]/(c-0)=f'(m)=[g(c)-g(0)]/(c-0)=g'(x) 0
绛旓細绗涓棰樼殑绛旀鏄0 鈭玣(x)dx涓庣Н鍒嗗彉閲忔槸鏃犲叧鐨勶紝涔熷氨鏄杩欓噷鐨剎鍙互鏀规垚浠绘剰鍙橀噺锛屾墍浠ヨ鈭玣(x)dx涓庘埆f(t)dt鏄竴涓剰鎬濓紝杩欓噷鐨勭Н鍒嗕笌f(t)dt鏄惁杩炵画鏄病鏈夊叧绯荤殑銆傛湜閲囩撼o!!!
绛旓細+f''(c1)(0锛峹)^2/2,f(1)锛漟(x)+f'(x)(1锛峹)+f''(c2)(1锛峹)^2/2.涓ゅ紡鐩稿噺,寰 f'(x)锛漟''(c1)x^2/2锛峟''(c2)(1锛峹)^2/2,鍙栫粷瀵瑰煎苟鍒╃敤鏉′欢寰 |f'(x)|<=M/2(x^2+(1锛峹)^2)<=M/2.鏈鍚庣殑涓嶇瓑寮忔槸鍥犱负x^2+(1锛峹)^2鍦╗0,1]涓婄殑鏈澶у兼槸1.
绛旓細浠(x)=f(x)-x^2,G(x)鍦紙0,1锛夊唴鍙锛孏(0)=f(0)锛孏(1)=f(1)-1=f(0)=G(0),鏈夋媺鏍兼湕鏃ュ畾鐞嗗緱鍦紙0,1锛夊唴瀛樺湪蠄锛屼娇寰桮'(蠄)=f'(蠄)-2蠄=0,鍗砯'(蠄)=2蠄
绛旓細璁 g(x)=f(x)*x^3 鍒欐湁锛歡'(x)=f(x)*3*x^2+f'(x)*x^3 鍥犱负锛歡(0)=g(a)=0 鏍规嵁涓煎畾鐞锛屽湪(0,a)涓瓨鍦ㄎ句娇寰梘'(尉)=0 鍗筹細f(尉)*3*尉^2+f'(尉)*尉^3=0 鎵浠ワ細f(尉)*3+f'(尉)*尉=0
绛旓細鍙堝洜涓烘槸鍗曡皟閫掑噺,瀹氱Н鍒哸~1灏忔柤f(a)*(1-a)鎵浠ユ垜浠湁涓や釜寮忓瓙 宸﹀紡涓轰粠0~a鐨勫畾绉垎 鍙冲紡涓篴*(0~a+f(a)*(1-a))鎶婁袱渚у悓鍑廰*(0~a)宸﹀紡鍙樻垚(0~a)*(1-a)鍙冲紡鍙樻垚f(a)*(1-a)*a 鍚岄櫎(1-a),姝ら」鎭掓鎵浠ヤ笉鐢ㄥ彉鍙 宸﹀紡鍙樻垚(0~a)鍙冲紡鍙樻垚f(a)*a 鍦ㄧ敱涓棿鍊煎畾鐞嗗彲鐭...
