f(x)=ï¼a+1)lnx+ax^2+1
å¾å°å®ä¹åï¼x>0
æ±å¯¼ï¼fâ(x)=(a+1)/ x+2ax
å½aâ¥0æ¶ï¼fâ(x) >0,åf(x)åè°éå¢
å½aâ¤-1æ¶ï¼fâ(x) <0ï¼åf(x)åè°éå
å½-1<a<0æ¶ï¼
设g(x)=xfâ(x)=2ax^2+a+1,
âµx>0;â´g(x)åfâ(x)åå·ã
æ¤æ¶å½xâ¥â(-(a+1)/2a)æ¶ï¼g(x)â¥0ï¼åfâ(x)â¥0ï¼é£ä¹f(x)åè°éå¢
æ¤æ¶å½0<x<â(-(a+1)/2a)æ¶ï¼g(x)<0ï¼åfâ(x)<0ï¼é£ä¹f(x)åè°éå.
å½åè°éåæ¶ï¼ææ大å¼1
f'(x)=(a+1)/x+2ax=(a+1+2ax^2ï¼/x,
对任æx1,x2âï¼0ï¼âï¼ï¼é½æ|f(x1)-f(x2)|â¥4|x1-x2|ï¼
â´|[f(x1)-f(x2)]/(x1-x2)|>=4,
â´|f'(x)|>=4,x>0,
â´|a+1+2ax^2|>=4x,
â´a+1+2ax^2>=4x,æa+1+2ax^2<=-4x,
â´a>=(4x-1)/(2x^2+1),æa<=-(4x+1)/(2x^2+1).
设g(x)=(4x-1)/(2x^2+1),å
g'(x)=[4(2x^2+1)-4x(4x-1)]/(2x^2+1)^2
=[4+4x-8x^2]/(2x^2+1)^2
=4(1-x)(1+2x)/(2x^2+1),
0<x<1æ¶g'(x)>0,g(x)âï¼x>1æ¶g'(x)<0,g(x)âã
â´g(x)|max=g(1)=1.
设h(x)=(4x+1)/(2x^2+1),å
h'(x)=[4(2x^2+1ï¼-4xï¼4x+1)]/(2x^2+1)^2
=4(1-x-2x^2)/(2x^2+1)^2
=4(1+x)(1-2x)/(2x^2+1)^2,
仿ä¸ï¼h(x)|max=h(1/2)=2,
â´a>=1,æa<=-2.
å 为a<-1,æ
æa<=-2
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解ï¼åå½æ°f(x)=(a+1)lnx+ax^2+1 ï¼å·²ç¥ï¼a<-1ï¼ä¸å¦¨è®¾x1>x2,ä¸x>0ã
åå½æ°ç导å½æ°f'(x)=(a+1)/x +2axãå 为a<-1,x>0 å¾ï¼
f'(x)<0ï¼æ以åå½æ°ä¸ºåå½æ°ï¼å³f(x2)-f(x1)>0
对äºä¸çå¼|f(x1)-f(x2)|>=4|x1-x2|æ¥è¯´ï¼ä»å ä½çæä¹æ¥ç解ï¼å°±æ¯å¨xçå®ä¹åéï¼å½æ°å¨
ç¹x2ä¸å线çæçå°äºçäº-4ãç»å¯¹å¼çåæ¢å¦ä¸ï¼
|f(x1)-f(x2)|/|x1-x2|>=4,å³ï¼[f(x1)-f(x2)]/(x1-x2)<=-4
æ ¹æ®å¯¼æ°çå®ä¹ï¼ä»¥ä¸ä¸çå¼çå«ä¹ä¸ºï¼å¨xå®ä¹åéï¼å¯¹åå½æ°ç导å½æ°çå¼åºå°äºçäº-4ã
å¾ï¼f'(x)<=-4, å³ (a+1)/x+2ax<=-4,åç®å¾ï¼
2ax^2+4x+(a+1)<=0
ç±äºa<0, 令u=2ax^2+4x+(a+1), åå½æ°u æ¯å¼å£åä¸çäºæ¬¡æç©çº¿å½æ°ï¼å½x=-1/aæ¶ï¼å½æ°uå
å°æ大å¼ï¼ï¼ä¹æ¯æ大å¼ï¼ãç±å¤å«å¼å¾ï¼4^2-4*2a*(a+1)<=0,ä¸çå¼ 2ax^2+4x+(a+1)<=0
ææç«ï¼å³ï¼2-a^2-a<=0 , å³ï¼a+2)(a-1)>=0,
æ以æï¼a<=-2,æa>=1ãa>=1ä¸å·²ç¥æ¡ä»¶ä¸ç¬¦ï¼èå»ã
综å以ä¸ï¼açåå¼èå´ä¸ºï¼a<=-2ã
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