※ 引述《victor7935 (victor)》之铭言： : 1. : A number c is called a fixed point of f if f(c)=c. : Prove that if f is differentiable on an interval I and f'(x)< 1 : for all x属於I,then f has at most one fixed point in I. : 2. : Show that the equation x^n+ax+b = 0, : n an even positive intefer, has at most two distinct real roots. : 谢谢︿︿ 1. 假设c1=f(c1), c2=f(c2) 因为根据Rolle's Theorem 存在c3在c1 c2之间使得f'(c3)=[f(c1)-f(c2)]/[c1-c2]=0 矛盾 所以fixed point最多只有一个 2. Let f(x)=x^n+ax+b f'(x)=nx^(n-1) +a=0 时 x^(n-1)=-a/n 只有一个实数值满足 所以f(x)只有一个极值 如果f(x)有三个实数解x1 x2 x3 f'(c1)=[f(x1)-f(x2)]/[x1-x2] =0 c1在x1 x2之间 f'(c2)=[f(x2)-f(x3)]/[x2-x3] =0 c2在x2 x3之间 表示f(x)有至少两个极值(矛盾) 所以f(x)最多只有两个相异实数解 --

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