[Everyday Mathematics]20150225

设 $f:\bbR\to\bbR$ 二次可微, 适合 $f(0)=0$. 试证: $$\bex \exists\ \xi\in\sex{-\frac{\pi}{2},\frac{\pi}{2}},\st f''(\xi)=f(\xi)(1+2\tan^2\xi). \eex$$

[Everyday Mathematics]20150225相关推荐

[Everyday Mathematics]20150222
设 $$\bex a_0=1,\quad a_1=\frac{1}{2},\quad a_{n+1}=\frac{na_n^2}{1+(n+1)a_n}\ (n\geq 1). \eex$$ 试证: ...
[Everyday Mathematics]20150204
设 $k_0>0$, $\phi:[k_0,\infty)\to[0,\infty)$ 是有界递减函数, 并且 $$\bex \phi(k)\leq \frac{A}{(k-h)^\al}\ph ...
[Everyday Mathematics]20150107
设 $f\in C^1[a,b]$, $f(a)=0$, 且存在 $\lm>0$, 使得 $$\bex |f'(x)|\leq \lm |f(x)|,\quad \forall\ x\in [a ...
[Everyday Mathematics]20150214
设 $\dps{x\in \sex{0,\frac{\pi}{2}}}$, 试比较 $\tan(\sin x)$ 和 $\sin(\tan x)$. 转载于:https://www.cnblogs.c ...
[Everyday Mathematics]20150203
设 $f$ 在 $\bbR$ 上连续可导, 且 $\dps{f'\sex{\frac{1}{2}}=0}$. 试证: $$\bex \exists\ \xi\in \sex{0,\frac{1}{2} ...
[Everyday Mathematics]20150103
试求极限$$\bex \vlm{n} \sez{\int_1^{e^2}\sex{\frac{\ln x}{x}}^n\rd x}^\frac{1}{n}.\eex$$ 转载于:https://www ...
[Everyday Mathematics]20150104
设 $a>0$, $$\bex x_1=1,\quad x_{n+1}=x_n+an\prod_{i=1}^n x_i^{-\frac{1}{n}}. \eex$$ 试证: $$\bex \vl ...
[Everyday Mathematics]20150221
设 $y_n=x_n^2$ 如下归纳定义: $$\bex x_1=\sqrt{5},\quad x_{n+1}=x_n^2-2\ (n=1,2,\cdots). \eex$$ 试求 $\dps{\vl ...
[Everyday Mathematics]20150113
设 $f\in C^2(0,+\infty)$ 适合 $$\bex \lim_{x\to 0^+}f'(x)=-\infty,\quad \lim_{x\to 0^+}f''(x)=+\infty. ...