当x∈(0,π/2)时,证明：x/(1+x²)＜arctanx＜x,

题目

当x∈(0,π/2)时,证明：x/(1+x²)＜arctanx＜x,

答案

arctanx导数=1/(1+x^2)x/(1+x^2)导数=(1-x^2)/(1+x^2)^2.令f(x)=arctanx-x/(1+x^2)则f(x)‘=1/(1+x^2)-(1-x^2)/(1+x^2)^2=2x^2/(1+x^2)^2在x∈(0,π/2)时f(x)>0,单调增,又因为f(0)=0,所以f(x)>0,即arctanx-x/(1+x^2)...