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Maxima and minima
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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