关于泰勒公式的专题讨论

$\bf命题：$$\bf(Landau不等式)$设$f(x)$在$\left( { - \infty , + \infty } \right)$上二次可微，且${M_k} = \mathop {Sup}\limits_{x \in \left( { - \infty , + \infty } \right)} \left| {{f^{\left( k \right)}}\left( x \right)} \right| < + \infty ,k = 0,2$

证明：${M_1} = \mathop {Sup}\limits_{x \in \left( { - \infty , + \infty } \right)} \left| {f'\left( x \right)} \right| < + \infty $，且${M_1} \le \sqrt {2{M_0}{M_2}} $

1

$\bf命题：$ 设$f\left( x \right) \in {C^2}\left[ {a,b} \right]$，且$\left| {\int_a^b {f\left( x \right)dx} } \right| < \int_a^b {\left| {f\left( x \right)} \right|dx} $，${M_1} = \mathop {Sup}\limits_{x \in \left[ {a,b} \right]} \left| {f'\left( x \right)} \right|,{M_2} = \mathop {Sup}\limits_{x \in \left[ {a,b} \right]} \left| {f''\left( x \right)} \right|$，则\[\left| {\int_a^b {f'\left( x \right)dx} } \right| \le \frac{{{M_1}}}{2}{\left( {b - a} \right)^2} + \frac{{{M_2}}}{6}{\left( {b - a} \right)^3}\]

1

$\bf命题：$设$f(x)$在$(-1,1)$上二阶可导，且$f\left( 0 \right) = f'\left( 0 \right) = 0$，若$$\left| {f''\left( x \right)} \right| \le \left| {f\left( x \right)} \right| + \left| {f'\left( x \right)} \right|$$在$(-1,1)$上总成立，则存在$x=0$的某个邻域内$f\left( x \right) \equiv 0$

1

$\bf命题：$设$f(x)$在$[0,1]$上二阶可导，且$f\left( 0 \right) = 0,f\left( 1 \right) = 3,\min \limits_{x \in \left[ {0,1} \right]} f\left( x \right) = - 1$，则存在$c\in (0,1)$，使得$f''\left( c \right) \ge 18$

1

$\bf命题：$设$f(x)$在$[-1,1]$上二阶可导，$f(0)=0$，则存在$\xi \in [-1,1]$，使得$f''\left( \xi \right) = 3\int_{ - 1}^1 {f\left( x \right)dx} $

1

$\bf命题：$设$f(x)$在$[a,b]$上二次可微，且$f'\left( {\frac{{a + b}}{2}} \right) = 0$，证明：

(1)存在$\xi \in(a,b)$，使得\[\left| {f''\left( \xi \right)} \right| \geqslant \frac{4}{{{{\left( {b - a} \right)}^2}}}\left| {f\left( b \right) - f\left( a \right)} \right|\]

(2)若$f(x)$不恒为常数，则存在$\eta \in(a,b)$，使得\[\left| {f''\left( \eta \right)} \right| > \frac{4}{{{{\left( {b - a} \right)}^2}}}\left| {f\left( b \right) - f\left( a \right)} \right|\]

1

$\bf(02中南七)$设$f(x)$在$\left( { - \infty , + \infty } \right)$上存在三阶导数${f^{\left( 3 \right)}}\left( x \right)$，且$f\left( { - 1} \right) = 0,f\left( 1 \right) = 1,{f^{\left( 1 \right)}}\left( 0 \right) = 0$，证明：$\mathop {Sup}\limits_{x \in \left( { - 1,1} \right)} {f^{\left( 3 \right)}}\left( x \right) \geqslant 3$