# A number a is called a fixed point of a function f if f(a)=a. Consider the function f(x)=x^87+4x+2, x in R. (a) Use the Mean Value Theorem to show that f(x) cannot have more than one fixed point. (b) Use the Intermediate Value Theorem and the result in (a) to show that f(x) has exactly one fixed point.

mikioneliir 2022-09-26 Answered
A number $a$ is called a fixed point of a function $f$ if $f\left(a\right)=a$. Consider the function $f\left(x\right)={x}^{87}+4x+2$, $x\in \mathbb{R}$.
(a) Use the Mean Value Theorem to show that $f\left(x\right)$ cannot have more than one fixed point.
(b) Use the Intermediate Value Theorem and the result in (a) to show that $f\left(x\right)$ has exactly one fixed point.
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## Answers (2)

Ashlynn Delacruz
Answered 2022-09-27 Author has 9 answers
(a) If $f$ has two distinc fixed points, namely $a, then
$f\left(b\right)-f\left(a\right)={f}^{\prime }\left(c\right)\left(b-a\right)$
for some $c\in \left(a,b\right)$. Then ${f}^{\prime }\left(c\right)=1$. But ${f}^{\prime }\left(x\right)={x}^{86}+4\ge 4$.
(b)Let $F\left(x\right)=f\left(x\right)-x$, which is continuous. $F\left(0\right)=2$ and $F\left(-1\right)=-2$. So there is $c\in \left(-1,0\right)$ such that $F\left(c\right)=0$ and, hence, $f\left(c\right)=c$. By (a), this is the only possible fixed point.
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gemauert79
Answered 2022-09-28 Author has 2 answers
$f\left(x\right)-x$ is an increasing function. It is positive at $x=0$ and negative at $x=-1$.
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