Extrema and Critical Points

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Extrema and Critical Points

The concepts of extrema and critical points of functions is an extremely useful concept in mathematics, particularly the areas of calculus and differential equations. Extrema may be absolute, such as the maximum (or highest) value or the minimum (or lowest) value attained by the function over its defined domain, or extrema may be local, such as the maximum or minimum value attained by a function in a particular, localized interval of the domain. Critical points are any interior points of the domain at which the derivative of the function is either zero or undefined (that is, does not exist). Importantly, on all closed intervals (that is, finite intervals that include boundary points), continuous functions will take both an absolute maximum and an absolute minimum; it is impossible to construct a continuous function over a closed interval that does not take on absolute maximum and minimum values.

Defining Absolute and Local Extrema

The definitions of absolute maximum and minimum values, as well as local maximum and minimum values, is exactly as expected:

• The absolute maximum is the maximum value of the function f across its entire domain D (that is, f(c) is the absolute maximum if f(c) >= f(x) for all x in D)
• Theabsolute minimum us the minimum value of the function across its entire domain (that is, f(c) is the absolute minimum if f(c) < = f(x) for all x in D)
• A local maximum is the maximum value of the function in a particular interval (that is, f(c) is a local maximum if f(c) > = f(x) for all x in a defined interval, which must contain c)
• A local minimum is the minimum value of the function in a particular interval (that is, f(c) is a local minimum if f(c) < = f(x) for all x in a defined interval, which must contain c)

Finding Absolute Extrema on Closed Intervals

To find an absolute extreme value of a function f on a closed interval, it is necessary to first find the critical points by solving f^' = 0 (that is, setting the first derivative of the function to zero and finding all solutions to this problem). Then, one evaluates the function f at all critical points, as well as the endpoints (that is, boundary points) of the closed interval. The absolute maximum of the function on the interval is the largest of all these values, and the absolute minimum of the function is the smallest of all these values.

Using the First Derivative to Find Local Extrema

Since the first derivative of a function provides the slope, or instantaneous rate of change, of the function at any given point, evaluating and comparing the signs of the first derivative on both "sides" of a critical point establishes whether a critical point is a local extreme value. If f^', the first derivative is positive, then the function is increasing. If f^' is negative, then the function is decreasing. A local maximum occurs when f^' changes from positive to negative at the critical value c (that is, f^'(x) > 0 for x < c and f^'(x) < 0 for x > c). A local minimum occurs when f^' changes from negative to positive at the critical value c (that is, f^'(x) < 0 for x < c and f^'(x) > 0 for x > c). If f^' does not change sign at the critical point c, then no local maximum or local minimum occurs at this point. This practice is sometimes known as the first derivative test.

Using the Second Derivative to Find Local Extrema

Interestingly enough, the second derivative can also be used to find local extreme values. The second derivative of a function, of course, is the derivative of the first derivative of the function. The second derivative of a function is often represented by the mathematical symbol f^''. The second derivative of a function provides insight into its concavity--over intervals where f^'' < 0, the function is concave down, and over intervals where f^'' > 0, the function is concave up. When the concavity of a function changes at a critical point, either a local maximum or local minimum is observed. More specifically, if f^'(c) = 0 and f^''(c) < 0, the function has a local maximum at x = c. Similarly, f^'(c) = 0 and f^''(c) > 0, the function has a local minimum at x = c. This evaluation is sometimes called the second derivative test.