已知函數(shù)f(x)=x3-3ax(a∈R),若直線(xiàn)x+y+m=0對(duì)任意的m∈R都不是曲線(xiàn)y=f(x)的切線(xiàn),則a的取值范圍為 .
【答案】
分析:首先分析對(duì)任意的m直線(xiàn)x+y+m=0都不是曲線(xiàn)y=f(x)的切線(xiàn)的含義,即可求出函數(shù)f(x)=x
3-3ax(a∈R)的導(dǎo)函數(shù),使直線(xiàn)與其不相交即可.
解答:解:f(x)=x
3-3ax(a∈R),則f(x)
′=3x
2-3a
若直線(xiàn)x+y+m=0對(duì)任意的m∈R都不是曲線(xiàn)y=f(x)的切線(xiàn),則直線(xiàn)的斜率為-1,f(x)
′=3x
2-3a與直線(xiàn)x+y+m=0沒(méi)有交點(diǎn),
又拋物線(xiàn)開(kāi)口向上則必在直線(xiàn)上面,即最小值大于直線(xiàn)斜率,
則當(dāng)x=0時(shí)取最大值,-3a>-1,
則a的取值范圍為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180327162878337/SYS201310241803271628783007_DA/0.png)
即答案為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180327162878337/SYS201310241803271628783007_DA/1.png)
.
點(diǎn)評(píng):此題只要考查函數(shù)與方程的綜合應(yīng)用,以及函數(shù)導(dǎo)函數(shù)的計(jì)算,屬于綜合性問(wèn)題,計(jì)算量小但有一定的難度,屬于中等題.