$f\circ f$ attains its minimum at at least two points

Question

Let $f$ be continuous on $[a,+\infty)$. Assume $\lim_{x\to+\infty}f(x)=+\infty$, and $f$ attain its minimum at $c>a$. Assume further $a\leq f(c)<c<f(a)$. How to prove that $f\circ f$ attains its minimum at at least two points? What extra function shall we construct then?

Answer 1

Note that $f\circ f$ is defined on $[a,\infty),$ $\lim_{x\to\infty} f\circ f(x) = \infty$ and it attains its minimum at points $b$ with $f(b)=c.$ So you need to show that $f$ obtains the value $c$ twice. Does this help?