一道定积分的不等式证明题
问题描述：
一道定积分的不等式证明题
设Pn(x)为n次多项式,求证：
∫(a,b)|Pn'(x)|dx
数学人气：334 ℃时间：2020-04-06 19:21:27
优质解答
设a≤x1
=|Pn(x1)-Pn(a)|+|Pn(x2)-Pn(x1)|+...+|Pn(xk)-Pn(b)|
≤|Pn(x1)|+|Pn(a)|+|Pn(x2)|+|Pn(x1)|+...+|Pn(xk)|+|Pn(b)|
≤Max|Pn(x)|+Max|Pn(x)|+...+Max|Pn(x)|=2k*Max|Pn(x)|
≤2n*Max|Pn(x)|
=|Pn(x1)-Pn(a)|+|Pn(x2)-Pn(x1)|+...+|Pn(xk)-Pn(b)|
≤|Pn(x1)|+|Pn(a)|+|Pn(x2)|+|Pn(x1)|+...+|Pn(xk)|+|Pn(b)|
≤Max|Pn(x)|+Max|Pn(x)|+...+Max|Pn(x)|=2k*Max|Pn(x)|
≤2n*Max|Pn(x)|
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答
设a≤x1
=|Pn(x1)-Pn(a)|+|Pn(x2)-Pn(x1)|+...+|Pn(xk)-Pn(b)|
≤|Pn(x1)|+|Pn(a)|+|Pn(x2)|+|Pn(x1)|+...+|Pn(xk)|+|Pn(b)|
≤Max|Pn(x)|+Max|Pn(x)|+...+Max|Pn(x)|=2k*Max|Pn(x)|
≤2n*Max|Pn(x)|