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 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.Finding maxima and minima