（1)设,证明:

设0＜λ＜1,<img src='https://img2.soutiyun.com/ask/2020-12-15/976886971942264.png' />,证明: <img src='https://img2.soutiyun.com/ask/2020-12-15/976887004344117.png' />

设f(x)在区间[0,1]上连续，证明:<img src='https://img2.soutiyun.com/ask/2020-12-15/976873792952469.png' />

设方程1+xy=c(x-y)[其中c为常数].证明:<img src='https://img2.soutiyun.com/ask/2021-01-10/979121448557899.png' />

设<img src='https://img2.soutiyun.com/ask/2020-12-17/977061005028657.png' />(n=3,4,5.....)，证明: (1)级数绝对收敛; (2)数列{a_n}收敛.

设a＞0，a≠1，证明：<img src='https://img2.soutiyun.com/ask/2021-01-18/979827996087283.png' />不存在，且不为无穷大，

设<img src='https://img2.soutiyun.com/ask/2020-08-07/965670498028166.png' />都是集合,其中n≥1.证明:如果 <img src='https://img2.soutiyun.com/ask/2020-08-07/96567052187253.png' /> 则 <img src='https://img2.soutiyun.com/ask/2020-08-07/965670540210579.png' />

设<img src='https://img2.soutiyun.com/ask/2020-11-27/975351283558174.png' />证明: (1)<img src='https://img2.soutiyun.com/ask/2020-11-27/975351295964883.png' /> (2)若<img src='https://img2.soutiyun.com/ask/2020-11-27/9753513102387.png' />

设<img src='https://img2.soutiyun.com/ask/2020-08-18/966607687193762.png' />证明多项式 <img src='https://img2.soutiyun.com/ask/2020-08-18/966607701896603.png' /> 在(0，1)内至少有一个零点.

设矩阵<img src='https://img2.soutiyun.com/ask/2020-12-28/978034148086704.png' />证明A可逆，并求A^-1。

设(f(x)=ln(1+x),x∈(-1,1).由拉格朗日中值定理得:<img src='https://img2.soutiyun.com/ask/2021-01-13/97939236664681.png' />.<img src='https://img2.soutiyun.com/ask/2021-01-13/979392378464486.png' />使得ln(1+x)-In(1+0)=<img src='https://img2.soutiyun.com/ask/2021-01-13/979392393557349.png' />证明:<img src='https://img2.soutiyun.com/ask/2021-01-13/979392405881054.png' />

(1)<img src='https://img2.soutiyun.com/shangxueba/ask/51243001-51246000/51244816/974391435559486.png' />(k≥2为正整数);(2)<img src='https://img2.soutiyun.com/shangxueba/ask/51243001-51246000/51244816/974391445908077.png' />

设函数f在点x=1处二阶可导,证明:若f'(1)=0,f"(1)=0,则在x=1处有<img src='https://img2.soutiyun.com/ask/2020-11-28/975441569605878.png' /> <img src='https://img2.soutiyun.com/ask/2020-11-28/97544157767434.png' />

设矩阵<img src='https://img2.soutiyun.com/ask/2020-12-16/976985818130937.png' />证明 (1)<img src='https://img2.soutiyun.com/ask/2020-12-16/976985833829835.png' />的充分必要条件是<img src='https://img2.soutiyun.com/ask/2020-12-16/976985850216772.png' />： (2)当<img src='https://img2.soutiyun.com/ask/2020-12-16/976985874079137.png' />时，A是不可逆矩阵

设函数<img src='https://img2.soutiyun.com/ask/2020-12-16/976976603992918.png' />,其中函数g(x)在(-∞,+∞)上连续,且 g(1)=5,<img src='https://img2.soutiyun.com/ask/2020-12-16/976976616554637.png' />,证明<img src='https://img2.soutiyun.com/ask/2020-12-16/976976676821084.png' />,并计算f''(1)和F'''(1).

设<img src='https://img2.soutiyun.com/ask/2021-01-28/980692750486118.png' /> (1)证明f(x)在[0,+∞)上可导，且一致连续; (2)证明反常积分<img src='https://img2.soutiyun.com/ask/2021-01-28/980692795149672.png' />发散。

设fe(x)可导，且fk(x)≠0，k=1，2，....，n， <img src='https://img2.soutiyun.com/ask/2021-01-18/979833162387778.png' />证明：<img src='https://img2.soutiyun.com/ask/2021-01-18/97983317623057.png' />

设函数f(x)在[01]上二阶可导,且f"(x)≤0,x∈[0,1],证明: <img src='https://img2.soutiyun.com/ask/2020-12-16/976976979900419.png' />

设一元函数f(u)在[-1,1]上连续，证明 <img src='https://img2.soutiyun.com/ask/2021-02-01/981026749760827.png' /> 其中Ω为单位球<img src='https://img2.soutiyun.com/ask/2021-02-01/981026759890406.png' />。

设f(x)=1/(a-x)，证明：<img src='https://img2.soutiyun.com/ask/2020-11-27/975320290099449.jpg' /> 而且 <img src='https://img2.soutiyun.com/ask/2020-11-27/975320308867523.jpg' />