Maxima and minima

ja:極値

In mathematics, a point x is a local maximum of a function f if there exists some ε > 0 such that f(x) ≥ f(c) for all c with |x - c| < 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(c) for all c with |x - c| < 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(c) for all c. Similarly, a global minimum is a point x for which f(x) ≤ f(c) for all c. 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. Either of those can be called an absolute extremum.

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.

If the function is defined over a bounded segment, one also need to check the end points of the segment.

Examples

• The function <math>x^2<math> has a unique global minimum at x = 0.
• The function <math>x^3<math> has no global or local minima or maxima. Although the first derivative (3<math>x^2<math>) is 0 at x = 0, the second derivative (6x) is also 0.
• The function <math>x^3<math>/ 3 - x has first derivative <math>x^2 - 1<math> 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.
• The function <math> x^3 + 3 x^2 - 2 x + 1 <math> defined over the closed interval (segment) [-4,2] (see graph) has two extremums: one local maximum in <math> x = (-1-\sqrt{15})/3 <math> , one local minimum in <math> x = (-1+\sqrt{15})/3 <math>, a global maximum on x=2 and a global minimum on x=-4.

Functions of more variables

For functions of more variables similar concepts apply, but there is also the saddle point.

See also

• Mechanical equilibrium