Question

Let y=f(x) be a continuous function defined on the closed interval [0,b] with the property that 0<f(x)<b for all x in [0,b]. Show that there exists a point c in (0,b) with the property that f(c)=c. How would set this up?

Answer 1

Please see below.

Explanation:

Let `g(x) = f(x)-x`

Note that `g` is also continuous on `[0,b]`.

Since `0 < f(0)`, we see that `g(0) > 0`

Since `f(b) < b`, we see that `g(b) < 0`

Therefore, by the Intermediate Value Theorem, there is a `c` in `(0,b)` with `g(c) = f(c) -c = 0` so that `f(c) = c`