题目

设 $f\left( x\right)$ 在 $\left\lbrack {0,1}\right\rbrack$ 上有二阶导数,且 ${f}^{\prime }\left( x\right) > 0,{f}^{\prime \prime }\left( x\right) > 0,f\left( 0\right) = 0$ . 取 ${x}_{1} \in \left( {0,1}\right)$ ,数列 $\left\{x_{n}\right\}$ 满足 $\left( {{x}_{n + 1} - {x}_{n}}\right) {f}^{\prime }\left( {x}_{n}\right) + f\left( {x}_{n}\right) = 0\left( {n = 1,2,\cdots }\right)$ . 证明: $\mathop{\lim }\limits_{n \rightarrow \infty }{x}_{n}$ 存在,并求其值.