求解一道大一的高数题.
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求解一道大一的高数题.
![](http://img.wesiedu.com/upload/1/8c/18c44cb30ed219673e8253d19b0c2b6e.jpg)
![](http://img.wesiedu.com/upload/1/8c/18c44cb30ed219673e8253d19b0c2b6e.jpg)
![求解一道大一的高数题.](/uploads/image/z/15158259-27-9.jpg?t=%E6%B1%82%E8%A7%A3%E4%B8%80%E9%81%93%E5%A4%A7%E4%B8%80%E7%9A%84%E9%AB%98%E6%95%B0%E9%A2%98.)
转证:(1+x)*[ln(1+x)] >= arctanx,
f(x) = (1+x)*[ln(1+x)] - arctanx,
f(0) = 0,
f'(x) = ln(1+x) +1 - 1/(1+x*x) = ln(1+x) +(x*x)/(1+x*x) >0,
so f(x)严格单调递增,f(x)>=f(0),so (1+x)*[ln(1+x)] - arctanx>=0,
so (1+x)*[ln(1+x)] >=arctanx,
so ln(1+x) >=arctanx/(1+x)
f(x) = (1+x)*[ln(1+x)] - arctanx,
f(0) = 0,
f'(x) = ln(1+x) +1 - 1/(1+x*x) = ln(1+x) +(x*x)/(1+x*x) >0,
so f(x)严格单调递增,f(x)>=f(0),so (1+x)*[ln(1+x)] - arctanx>=0,
so (1+x)*[ln(1+x)] >=arctanx,
so ln(1+x) >=arctanx/(1+x)