Maxima and minima
In mathematics, a point x is a local maximum of a function f if there exists some ε > 0 such that f(x) ≥ f(y) for all y with |x - y| < ε. Stated less formally, a local maximum is a point where the function takes on its largest value among all points in the immediate vicinity. On a graph of a function, its local maxima will look like the tops of hills.A local minimum is a point x for which f(x) ≤ f(y) for all y with |x - y| < ε. On a graph of a function, its local minima will look like the bottoms of valleys.
A global maximum is a point x for which f(x) ≥ f(y) for all y. Similarly, a global minimum is a point x for which f(x) ≤ f(y) for all y. Any global maximum or (minimum) is also a local maximum (minimum); however, a local maximum or minimum need not also be a global maximum or minimum.
Finding maxima and minima
Finding global maxima and minima is the goal of optimization. For twice-differentiable functions in one variable, a simple technique for finding local maxima and minima is to look for stationary points, which are points where the first derivative is zero. If the second derivative at a stationary point is positive, the point is a local minimum; if it is negative, the point is a local maximum; if it is zero, further investigation is required.
Examples
- The function x² has a unique global minimum at x = 0.
- The function x³/3 - x has first derivative x² − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of the second derivative we can see that −1 is a local maximum and +1 is a local minimum. Note that this function has no global maxima or minima.
- The function |x| has a global minimum at x=0 that cannot be found by taking derivatives, because the derivative does not exist at x=0.
- The function cos(x) has infinitely many global maxima at 0, ±2π, ±π, ..., and infinitely many global minima at ±π, ±3π, ... .
- The function cos(x) - x has infinitely many local maxima and minima, but no global maxima or minima.