1,è¯æ:设x1âR,x2âR
[f(x1)+(x2)]-2[f(x1+x2)/2]=ax1^2+x1+ax2^2+x2-2a[(x1+x2)/2]^2-(x1+x2)=a[x1^2+x2^2-(x1+x2)^2/2]=a(x1-x2)^2/2
âµaï¼0,â´a(x1-x2)^2/2â¥0,å³1/2[f(x1)+f(x2)]â¥f[ï¼x1+x2)/2]
f(x)å¨Rä¸æ¯å¹å½æ°
2,è¥a=0,åf(x)=x,0â¤xâ¤1,|f(x)|â¤1æç«,æ¤ç§æ
åµç¬¦å
è¥aâ 0,åf(x)=ax^2+x,顶ç¹(-1/2a,-1/4a)
â å½-1/2aï¼0,å³a大äº0æ¶,满足|f(0)|â¤1,|f(1)|â¤1,解å¾-2â¤aâ¤0,ä¸aï¼0çç¾,æ¤æ
åµaæ 解
â¡å½-1/2aâ¥1,å³-1/2â¤aï¼0æ¶,满足|f(0)|â¤1,|f(1)|â¤1,解å¾-2â¤aâ¤0
â´-1/2â¤aï¼0
â¢å½0ï¼-1/2aï¼1,å³aï¼-1/2æ¶,满足|f(0)|â¤1,|f(1)|â¤1ä¸|-1/4a|â¤1,解å¾-2â¤aâ¤-1/4
â´-2â¤aï¼-1/2
ç±â â¡â¢å¾-2â¤aï¼0
综ä¸å¾açåå¼èå´ä¸º-2â¤aâ¤0
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