Concave, convex functions and points of inflection
Concave functions If f''(x) ≤ 0 in a given interval of x, the function is said to be concave. Convex functions If f''(x) ≥ 0 in a given interval of x, the function is said to be convex. Point of Inflection The point at which a curve changes being concave to convex or vice versa is called a point of inflection. E.g. f(x) = x 3 - 2x² + 5x - 4 f'(x) = 3x² - 4x + 5 f''(x) = 6x - 4 At the point of inflection, f''(x) = 0 6x - 4 = 0 x = 2/3 = 0.67 There is a point of inflection at x = 0.67. If x = 0.6 f''(0.6) = 3.6 - 4 = -0.4 f''(0.6) < 0 - the function is concave. If x = 0.8 f''(0.8) = 4.8 - 4 = 0.8 f''(0.8) > 0 - the function is convex. In the above simulation, there are two points of inflection. The simulation can be practised interactively here; just move the point gently to see the change. Points of inflection on the Bell Curve - Normal Distribution There are two points of inflection on the bell curve.