怎么证明|arctanX-arctanY|

题目

怎么证明|arctanX-arctanY|

答案

arctanx在实数范围内上连续且可导.
那么在内至少有一值c,使以下等式成立(拉格朗日中值定理)
arctanx-arctany=(arctan'c)(x-y)
arctanx-arctany=(1/(1+c²))(x-y)
(arctanx-arctany)/(x-y)=1/(1+c²)
又∵0<1/(1+c²)≤1 (c∈R)
∴0<(arctanx-arctany)/(x-y)≤1
∴|(arctanx-arctany)/(x+y)|<=1
|arctanx-arctany|<=|x-y|