Question

Let $f:R\to R$ be a continuous decreasing function. A point $x_0\in R$ is said to be a fixed point of $f$ if $f\left(x_0\right)=x_0.$
The number of fixed points of $f\circ f\circ f$ equals

• A. $0$
• B. $1$
• C. $2$
• D. infinitely many

Answer 1

The correct option is B $1$

Let $x,y\in D_f=R$ and $x<y$
$\Rightarrow f\left(x\right)\ge f\left(y\right)$
$\Rightarrow f\left(f\right(x\left)\right)\le f\left(f\right(y\left)\right)$
​​​​​​​​​​​​​​$\Rightarrow f\left(f\right(f\left(x\right)\left)\right)\ge f\left(f\right(f\left(y\right)\left)\right)$
​​​​​​​​​​​​​​​​​​​​​$\Rightarrow (f\circ f\circ f)\left(x\right)\ge (f\circ f\circ f)\left(y\right)$
​​​​​​​​​​​​​​​​​​​​​$\Rightarrow f\circ f\circ f$ is decreasing function on $R$.

​​​​​​​Therefore, it has only one fixed point.