# 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 has a unique global minimum at
*x*= 0. - The function has no global or local minima or maxima. Although the first derivative (3) is 0 at
*x*= 0, the second derivative (6*x*) is also 0. - The function / 3 - x has first derivative and second derivative 2
*x*. 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.