设fx为奇函数且在x=0处连续,证明f(0)=0 问题是证明fx在点x=0处连续。但怎么知道f（0）=0
\u82e5f\uff08x\uff09\u662f\u5947\u51fd\u6570\u4e14f'\uff080\uff09\u5b58\u5728\uff0c\u5219\u70b9x\uff1d0\u662f\u51fd\u6570F\uff08x\uff09\uff1df\uff08x\uff09/x\u7684\uff08\u8fde\u7eed\u70b9\uff09 \u600e\u6837\u8bc1\u660e\uff1f\u9996\u5148f(X)\u662f\u5947\u51fd\u6570\uff0cf\uff080\uff09=0\uff0c\u5219F\uff08x\uff09\u5728x=0\u70b9\u662f0/0\u578b\uff0c\u53ef\u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0cF'(0)=f'(0),\u4e14f'\uff080\uff09\u5b58\u5728\uff0c\u5219F\uff08x\uff09\u5728x=0\u5904\u53ef\u5bfc\uff0c\u53c8\u53ef\u5bfc\u5fc5\u8fde\u7eed\uff0c\u6545\u5f97\u8bc1
\u53f3\u8fb9\u4e0d\u7b49\u5f0f\u4ee4x=0\u5c31\u5f97\u52300\u2264f\uff080\uff09\u22640
因为是奇函数，所以有f（-x）=-f（x），且在x=0处有定义，所以显然是f（-0）=-f（0）；即2f（0）=0；f（0）=0绛旓細瑙ｇ敱褰搙锛0鏃讹紝fx=x²-x-2锛0 鍗筹紙x-2锛夛紙x+1锛夛紴0 鍗硏锛2鎴杧锛-1 鍗虫鏃秄锛坸锛夛紴0鐨勮В鏄痻锛-1 褰搙锛0鏃 -x锛0 鍗砯锛-x锛=锛-x锛塣2-锛-x锛-2=x^2+x-2 鍙堢敱f锛坸锛鏄鍑芥暟 鏁協锛坸锛=-x^2-x+2 鐢眆锛坸锛夛紴0 鍗-x^2-x+2锛0 鍗硏^2+x-2锛0 鍗...
绛旓細瀹屾暣鐨勮瘉鏄庤繃绋嬪鍥炬墍绀
绛旓細f(x)=ae^2x,x鈮0 f(x)=x+2, x>0 lim(x鈫0-)=a lim(x鈫0+)=2 鈭村綋a=2鏃讹紝x=0澶 宸︽瀬闄=鍙虫瀬闄=鍑芥暟鍊硷紝f(x)杩炵画銆
绛旓細1銆佸綋x>0鏃,f(x)=x³锛2x²锛1锛2銆佸綋x=0鏃,f(x)=0锛3銆佸綋x<0鏃,鍒欙紞x>0,鍥爁(x)鏄鍑芥暟,鍒檉(x)=锛峟(锛峹),鍥狅紞x>0,鍒檉(锛峹)=(锛峹)³锛2(锛峹)²锛1=锛峹³锛2x²锛1,鍒欏綋x<0鏃,f(x)=锛峹³锛2x²锛1 灏嗕笂杩颁笁娈...
绛旓細涓鏍 x<0,-x>0 鎵浠(-x)=-x|-x-2|=-x|x+2| 濂囧嚱鏁鍒檉(x)=-f(-x)=x|x+2| 涓攆(0)=0锛岀鍚堣繖涓紡瀛 鎵浠(x)= x|x+2|锛寈鈮0 x|x-2|銆倄>0
绛旓細璁緓<0锛岄偅涔-x>0銆傗埖x鈮0鏃讹紝f锛坸)=x^2-2x 鈭-x>0鏃舵湁f锛-x)=x^2+2x 鍙堚埖y=fx鏄瀹氫箟鍦≧涓婄殑濂囧嚱鏁 鈭磃锛-x)=-f锛坸锛=x^2+2x 鈭-f锛坸锛=x^2+2x 鈭磃锛坸锛=-x^2-2x 鈭磝<0鏃秄锛坸锛=-x^2-2x
绛旓細x=0 f(-0)=-f(0)f(0)=0 x<0 -x>0 f(-x)=(-x)^2+2x+3=x^2+2x+3=-f(x)鈭村垎娈鍑芥暟 f(x)=鈶爔^2-2x+3(x>0)鈶0(x=0)鈶-x^2-2x-3(x<0)
绛旓細鍥炵瓟锛璁綳銆=0,鍒-x銆0,f(-x)=-x(1+x)=-x-x^2 濂囧嚱鏁,f(-x)=-f(x) f(x)=x^2+x 绛旀灏辨槸杩欎釜浜,涓瀹氳璁板緱琛ㄥ畾涔夊煙
绛旓細f(x)鏄鍑芥暟 鎵浠(-x)=-f(x)鎵浠ュ綋x灏忎簬0鏃讹紝f(x)=-x(1+x)
绛旓細浠<0 -x>0 褰搙>0鏃,f(x)=x^2+sinx锛宖(-x)=(-x)^2+sin(-x)锛-f(x)=x^2-sinx锛 f(x)=-x^2+sinx锛宖(x)=x^2+sinx(x>0) f(x)=-x^2+sinx(x<0)
