In this chapter, we want to consider some classical minimization problems settled over suitable spaces of functions of one real variable, possibly vector-valued. Rather that aiming at giving the most complete theory, we focus on some particular examples, which are still quite instructive and interesting. In particular, in this chapter we treat the brachistocrone problem, the one-dimensional classical oscillator and give a proof of the celebrated isoperimetric inequality for smooth sets in the plane. We also discuss the exact determination of various sharp Poincaré constants for functions on an interval.