微积分(A)每日一题[39]

\[\begin{aligned} &F(x)=\int_0^{x}f(t)dt,F'(x)=f(x) \\ &F'(x)=f(x) \le \sqrt{1+2F(x)} \\ &\int_0^{x}\frac{F'(t)}{\sqrt{1+2F(t)}}dt \le \int_0^xdx=x \\ &\sqrt{1+2F(x)}-\sqrt{1+2F(0)}\le x \\ &f(x) \le \sqrt{1+2F(x)}\le \sqrt{1+2F(0)}+x=1+x \end{aligned} \]

\[H(x)=p(x)+p'(x)+\cdots+p^{(n)}(x) \\ H(x)=p(x)+H'(x) \Rightarrow H(x)-H'(x) = p(x) \ge 0 \\ \left( \frac{H(x)}{e^x} \right)' \le 0 \Rightarrow \frac{H(x)}{e^x} \ge \lim_{x \to +\infty}\frac{H(x)}{e^x}=0 \Rightarrow H(x) \ge 0 \]