Single-variable calculus from first principles
Convexity is the shape that makes optimisation easy. A convex function cups upward everywhere, like a bowl, and that one property makes it easy to minimise: there's exactly one lowest point, and any downhill path leads straight to it.
There are three equivalent ways to see convexity. First, the second derivative is non-negative everywhere: f″(x) ≥ 0. Second, the curve cups up and never bends downward. Third, the defining picture, a chord between any two points lies above the curve.
Picture a smooth valley, or the inside of a bowl, and drop a marble anywhere along it. No matter where it starts, the marble always rolls down to the single lowest point and settles there. That is exactly what convexity buys you: one valley, no false bottoms, so any downhill path leads to the one true minimum.