When an object is dropped from a great height, it not only falls but does so faster and faster (up to a certain limit). In other words, at our scale, its speed is an increasing function of time. Thus, if we plot distance travelled against time, we obtain a graph whose slope steadily increases and whose graph curves upward. This is the concept we seek to generalize and analyse mathematically.
Functions we vex ------------------------
A real-valued function defined on an interval is said to be convex if its rate of increase is itself an increasing function of the variable. If the function is differentiable, this amounts to saying that its derivative is increasing; if it is twice differentiable, that its second derivative is non-negative everywhere.
A quick glance at its graph shows that the line segment joining any two points on the graph lies above the portion of the graph between them. For differentiable functions, there is another characterization: the tangent at every point lies below the graph.
A function f is said to be concave if its negative, that is, –f, is convex. For a differentiable function, this amounts to saying that its derivative is decreasing or, if it is twice differentiable, that its second derivative is negative.