Metric Geometry determining fixed points

Question

Let $f\colon\mathbb{R}\to\mathbb{R}$ be given by $f(x)= e^{-x}$.

Show that $f$ has a fixed point and determine what it is.

Answer 1

$f(x) = x \iff e^{-x} = x \iff e^{-x} - x = 0$.

Consider $g(x) = e^{-x} - x$. We have: $g(0) = 1 > 0$, and $g(1) = e^{-1} -1 <0$. Since $g$ is continuous on $(0,1)$, MVT says there exists $c \in (0,1)$ such that $g(c) = 0 \Rightarrow e^{-c} - c = 0 \Rightarrow f(c) - c = 0 \Rightarrow f(c) = c \Rightarrow $ $c$ is the fixed point.