证明:函数在[0,+∞]连续.

设函数fz)在[0,1]上连续,在(0,1)内可导,且<img src='https://img2.soutiyun.com/ask/2020-12-27/977943646888836.png' />证明在(0,1)内存在一点c,使得f'(c)=0.

证明:(1)函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120252580493.png' />在原点(0,0)连续,但不存在偏导数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120271103552.png' />. (2)函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120281243132.png' />在原点(0,0)不连续[见题1(3)],但有偏导数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120383280969.png' />.

证明:若函数f(x)在[a,b]连续、非负,且<img src='https://img2.soutiyun.com/ask/2020-11-12/97406224084526.png' />使f(x0)＞0,则 <img src='https://img2.soutiyun.com/ask/2020-11-12/974062250390806.png' />

证明:若函数f(x)在a连续,则函数 <img src='https://img2.soutiyun.com/ask/2020-11-11/973957882099598.png' /> 在a都连续.

证明:若函数f(x)在[O,+∞)连续,且<img src='https://img2.soutiyun.com/ask/2020-11-12/974063888799517.png' />则<img src='https://img2.soutiyun.com/ask/2020-11-12/974063900815204.png' />

设f(x)为[0,1]上的非负单调非增连续函数(即当x＜y时,f(x)≥f(y)).利用积分中值 定理证明:对于0＜a＜β＜1.有下面的不等式成立 <img src='https://img2.soutiyun.com/ask/2020-11-30/975612485146551.png' />

设函数f(x)在[a,b]上连续,且f(x)＞0,证明:在(a,b)内存在一个ξ,使得 <img src='https://img2.soutiyun.com/ask/2021-01-14/979465674691464.png' />

设函数f(x)在[0，1]上连续，且f(0)= f(1)，证明一定存在x∈(0,<img src='https://img2.soutiyun.com/ask/2020-12-20/977320815878019.png' />)使得f(x₀)= f(x₀+<img src='https://img2.soutiyun.com/ask/2020-12-20/977320902712985.png' />).

设函数f(x)和g(x)在[0,1]上有连续导数,且f(0)=0,f'(x)≥0,g'(x)≥0.证明:对任何a∈[0,1],都有 <img src='https://img2.soutiyun.com/ask/2020-12-13/97672399961901.png' />

设ƒ (χ)在(-∞, +∞)内连续，且ƒ (χ)＞0.证明函数<img src='https://img2.soutiyun.com/ask/2020-08-30/967622834839226.png' />在(0,+ ∞)内为单调增加函数.

设函数f(u,v)在R²上具有二阶连续偏导数。证明:函数 <img src='https://img2.soutiyun.com/ask/2021-02-02/981107588858527.png' />

已知函数f(x)在[0，1]上连续，在(0，1)内可导，且f(0)＝0，f(1)＝1．证明： (I)存在ξ∈(0，1)，使得f(ξ)＝1－ξ； (Ⅱ)存在两个不同的点η，ζ∈(0，1)，使得fˊ(η)fˊ(ζ)＝1．

证明:函数<img src='https://img2.soutiyun.com/ask/2021-01-10/979120577570081.png' />在点(0,0)连续且有偏导数,但不可微分.

设函数f(x)在[0,1]上连续,在(0,1)内可导,且<img src='https://img2.soutiyun.com/ask/2020-08-07/965639441738848.png' />证明在(0,1)内存在一点ξ,使f'(ξ)=0。

证明:T函数在区间(0,+∞)存在任意阶连续导数,n∈N+,有 <img src='https://img2.soutiyun.com/ask/2020-11-13/974144501001748.png' />

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).

若连续函数列{φ₀(x)，φ₁(x)，…}在[a，b]上带权ρ(x)正交，且<img src='https://img2.soutiyun.com/ask/2020-07-31/965040632752842.png' />内恒正，证明:,对任意n个数<img src='https://img2.soutiyun.com/ask/2020-07-31/965040650416852.png' />，广义多项式<img src='https://img2.soutiyun.com/ask/2020-07-31/965040667564833.png' />在[a，b]上至少有一个零点。

证明:若函数f(x)在区间I连续,且对任意有理数x∈I,有f(x)=0,则<img src='https://img2.soutiyun.com/ask/2020-11-11/973956935040429.png' />