求曲线的凹凸区间和拐点求曲线y=e^arctanx的凹凸区间和拐点
问题描述：
求曲线的凹凸区间和拐点
求曲线y=e^arctanx的凹凸区间和拐点
答
y=e^arctanx
y'=[1/(1+x^2)]*e^arctanx
y''=[1/(1+x^2)]^2*e^arctanx
+(-2x)/(1+x^2)^2*e^arctanx
y''=0 得x=1/2
答
函数定义域为(-∞,∞)
y''=-2/(1+x^2)^2*exp(atan(x))*x+1/(1+x^2)^2*exp(atan(x))
令y''=0得x=1/2
观察y''在x=1/2两侧符号，在x0 x>1/2时y''所以凹区间x>1/2 凸区间x
答
y'=e^arctanx/(x^2+1)
y"
=[e^arctanx*(x^2+1)/(x^2+1)+e^arctanx*2x]/(x^2+1)^2
=e^arctanx*(1+2x)/(x^2+1)^2
y"=0得到x=-1/2
在x在x>-1/2时y">0,函数y为凹函数
x=-1/2时y=e^[-arctan(1/2)]
凸区间为(-无穷,-1/2)
凹区间为(-1/2,+无穷)
拐点(-1/2,e^[-arctan(1/2)])