Maxima Minima And Point Of Inflection

In mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighborhood (local extremum) or on the function domain in its entirety (absolute extremum). More generally, the maxima and minima of a set (as defined in set theory) are the greatest and least values in the set. To locate extreme values is the basic objective of optimization.

A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x*, if there exists some e > 0 such that f(x*) ≥ f(x) when |x - x*| < e. The value of the function at this point is called maximum of the function. Similarly, a function has a local minimum point at x*, if f(x*) ≤ f(x) when |x - x*| < e. The value of the function at this point is called minimum of the function.

A function has a absolute maximum point at x* if f(x*) ≥ f(x) for all x. Similarly, a function has a absolute minimum point at x* if f(x*) ≤ f(x) for all x. The absolute maximum and absolute minimum points are also known as the arg max and arg min: the argument (input) at which the maximum (respectively, minimum) occurs.

Say that f (x, y) has a critical point at (a, b) if and only if (a, b) = 0. (a, b) = 0

It is clear by comparison with the single variable result, that a necessary condition that f have a local extremum at (a, b) is that it have a critical point there, although that is not a sufficient condition. We refer to this as the first derivative test. We can get more information by looking at the second derivative. The second derivative test is employed to determine if a critical point is a relative maximum or a relative minimum. If f''(x_c)>0, then x_c is a relative minimum. If f''(x_c) > 0, then x_c is a maximum. If f''(x_c)=0, then the test gives no information. The notions of critical points and the second derivative test carry over to functions of two variables. Let z=f(x,y). Critical points are points in the xy-plane where the tangent plane is horizontal. Let f be the function defined in a neighborhood of a point a, and differentiate them. Let the curve y = f(x) be the graph of the function. Let f'(a) be finite, i.e., the tangent is not parallel to the y- axis. If the curve lies towards the positive direction of y axis we say the curve is concave upwards. If the curve is towards the negative direction of y axis, we say the curve is concave downwards The point that separates the convex part of a continuous curve from the concave part is called the point of inflection.