26.设\(a_1,\cdots,a_p\)为正数\((p\geqslant0)\),求极限\(\lim\limits_{x\rightarrow0+0}(\frac{a^x_1+a^x_2+\cdots+a^x_p}{p})^\frac{1}{x}\).
解\(\lim\limits_{x\rightarrow0+0}(\frac{a^x_1+a^x_2+\cdots+a^x_p}{p})^\frac{1}{x}=\exp(\lim\limits_{x\rightarrow0+0}\frac{\ln(\frac{a^x_1+a^x_2+\cdots+a^x_p}{p})}{x})\),
\(\lim\limits_{x\rightarrow0+0}\frac{\ln(\frac{a^x_1+a^x_2+\cdots+a^x_p}{p})}{x}=\lim\limits_{x\rightarrow0+0}\frac{\frac{a^x_1+a^x_2+\cdots+a^x_p}{p}-1}{x}=\sum\limits_{i=1}^p\lim\limits_{x\rightarrow0+0}\frac{a^x_i-1}{px}=\sum\limits_{i=1}^p\frac{\ln a_i}{p}\),
\(\lim\limits_{x\rightarrow0+0}(\frac{a^x_1+a^x_2+\cdots+a^x_p}{p})^\frac{1}{x}=\exp(\sum\limits_{i=1}^p\frac{\ln a_i}{p})=\sqrt[p]{a_1a_2\cdots a_p}\).
27.证明:
(1)三次方程\(x^3+x+1=0\)必有一实根;
证明 记\(f(x)=x^3+x+1,f(-1)=-1,f(0)=1\),则\(f(x)=0\)在\((-1,0)\)上必有一实根.
(2)方程\(\tan x-x=0\)有无穷多个实根.
证明 记\(f(x)=\tan x,f(2n\pi)=-2n\pi,f(2n\pi+\frac{\pi}{2}-0)=+\infty,\forall n\in\mathbb{N}\),则\(f(x)=0\)在\((2n\pi,2n\pi+\frac{\pi}{2})\)上必有一实根,故\(f(x)=0\)有无穷多个实根.
28.设函数\(f(x)\)在区间\(I\)上连续,且\(x_1,x_2,\cdots,x_n\)是\(I\)上任意\(n\)个点.证明:若对\(\forall x\in I\),有\(f(x)>0\),则存在\(\xi\in I\),使\(f(\xi)=\sqrt[n]{f(x_1)f(x_2)\cdots f(x_n)}\).
证明 由介值定理立得.
29.设函数\(f(x)\)在\((a,b)\)上连续,而且存在\(\{x_n\},\{y_n\}\subset(a,b)\),满足\(\lim\limits_{n\rightarrow\infty}x_n=b=\lim\limits_{n\rightarrow\infty}y_n\),使得
\[\lim\limits_{n\rightarrow\infty}f(y_n)=B>A=\lim\limits_{n\rightarrow\infty}f(x_n)\].
证明:对任意的\(\eta\)满足\(A<\eta<B\),必存在\(\{z_n\}\subset(a,b)\),满足\(\lim\limits_{n\rightarrow\infty}z_n=b\),使得\(\lim\limits_{n\rightarrow\infty}f(z_n)=\eta\).
证明 \(\exists N_1,\forall n>N_1,f(y_n)>\eta,\exists N_1,\forall n>N_1,f(x_n)<\eta\),
取\(N=max\{N_1,N_2\},\forall n,f(y_{N+n})>\eta>f(x_{N+n}),\)
必存在\(z_n\)在\(x_{N+n}\)和\(y_{N+n}\)之间,使得\(f(z_n)=\eta\),此时有\(\lim\limits_{n\rightarrow\infty}z_n=b\).
30.设函数\(f(x)\in C[a,b],\)且\(f([a,b])\subset[a,b]\),证明:存在\(c\in[a,b]\),使得\(f(c)=c\),即\(f(x)\)在\([a,b]\)上有不动点.
证明 记\(F(x)=f(x)-x,F(a)\geqslant0,F(b)\leqslant0\),由介值定理,存在\(c\in[a,b]\)使得\(F(c)=0\).
