Let $f: \mathbb R\to \mathbb R$ be a continuous function. Which of the following are sufficient conditions for $f$ to have a fixed point in $[0, 1]$?

Question

(a) $f(0)=f(1)$

(b)$f(1)<0<f(0)$

(c) $0<f(1)<f(0)$

(d) $f(0)<0<1<f(1)$

To obtain a fixed point, we should find $x=f(x)$ but how do I obtain the necessary conditions? What concept do I lack to begin solving this?

Answer 1

Hint: Use the intermediate value theorem on $f(x)-x$ which is also continuous.