Question

用langrange中值定理证明等式arctanx+arctan1/x=π/2

Answer 1

证明过程如下: 设f(x)=arctanx+arctan1/x 当x=1时,f(1)=arctan(1)+arctan(1)=π/2 当x=a,a>0且≠1 f'(x)=1/(1+x^2)+1/(1+(1/x)^2))*(-1/(x^2))=0 对任意a>0,f(x)在[a,1