Why a continuous function which goes $∞$ when $x→±∞$ has minimum value?
Solution 1:
Let $N$ be the minimal value of $f$ in $[0,1]$, which exists by compactness. Choose $M$ such that for $\lvert x \rvert > M$ we have $\lvert f(x) \rvert > N$. The minimum of $f$ is then attained in $[-M,M]$.