关于积分中值定理的专题讨论

$f命题：$设$fleft( x ight) in {C^1}left[ {0,1} ight]$，则存在$xi in (0,1)$，使得$int_0^1 {fleft( x ight)dx}  = fleft( 0 ight) + frac{1}{2}f'left( xi   ight)$

1

$f命题：$设$fleft( x ight) in Cleft[ {0,pi } ight],int_0^pi  {fleft( x ight)dx}  = 0,int_0^pi  {fleft( x ight)cos xdx}  = 0$，则存在${xi _1},{xi _2} in left( {0,pi } ight)$，使得$fleft( {{xi _1}} ight) = fleft( {{xi _2}} ight) = 0$

1

$f命题：$设$fleft( x ight) in Cleft[ {0,1} ight],fleft( x ight) > 0$，则对于$n in {N_ + }$，存在${xi _n}$，使得[frac{1}{n}int_0^1 {fleft( x ight)dx}  = int_0^{{xi _n}} {fleft( x ight)dx}  + int_{1 - {xi _n}}^1 {fleft( x ight)dx} ][mathop {lim }limits_{n o infty } n{xi _n} = int_0^1 {fleft( x ight)dx} /[fleft( 0 ight) + fleft( 1 ight)]]

1

$f命题：$设$fleft( x ight) in {C^1}left[ {0,1} ight],f'left( 0 ight) e 0$，则对$int_0^x {fleft( t ight)dt}  = fleft( {xi left( x ight)} ight)x,0 < x < 1$中的$xi(x)$，有$xi left( x ight)/x o 1/2left( {x o {0^ + }} ight)$

1

$f命题：$设$f(x)$在$[a,b]$上可积，证明：$$ lim limits_{n o infty } int_a^b {fleft( x ight)left| {sin nx} ight|dx}  = frac{2}{pi }int_a^b {fleft( x ight)dx} $$

1

$f命题：$设$f(x)$在$[-1,1]$上二阶连续可微，则存在$xi in(-1,1)$，使得[int_{ - 1}^1 {xfleft( x ight)dx}  = frac{2}{3}f'left( xi   ight) + frac{1}{3}xi f''left( xi   ight)]

1

$f命题：$设$phi(x)$是以$T$为周期的有界函数，且$frac{1}{T}int_0^T {varphi left( x ight)dx}  = C$，证明：[mathop {lim }limits_{n o infty } nint_n^{ + infty } {frac{{varphi left( t ight)}}{{{t^2}}}dt}  = C]

1

$f命题：$