question_answer

Suppose\[p\,(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+.....+{{a}_{n}}{{x}^{n}}\]. If\[|p\,(x)|\,\le \,|{{e}^{x}}^{-1}-1|\]for all \[x\ge 0,\] then ____.

A)  \[|{{a}_{1}}+2{{a}_{2}}+3{{a}_{3}}+...+n{{a}_{n}}|\le 1\]

B)  \[|{{a}_{1}}+2{{a}_{2}}+3{{a}_{3}}+...+n{{a}_{n}}|\ge 1\]

C)  \[|{{a}_{1}}+2{{a}_{2}}+3{{a}_{3}}+...+n{{a}_{n}}|\ge 0\]

D)  \[|{{a}_{1}}+2{{a}_{2}}+3{{a}_{3}}+...+n{{a}_{n}}|\le 0\]