Maxima and Minima

The terms maxima and minima refer to extreme values of a function, that is, the maximum and minimum values that the function attains. Maximum means upper bound or largest possible quantity. The absolute maximum of a function is the largest number contained in the range of the function. That is, if f(a) is greater than or equal to f(x), for all x in the domain of the function, then f(a) is the absolute maximum. For example, the function f(x) = -16x² + 32x + 6 has a maximum value of 22 occurring at x = 1. Every value of x produces a value of the function that is less than or equal to 22, hence, 22 is an absolute maximum. In terms of its graph, the absolute maximum of a function is the value of the function that corresponds to the highest point on the graph. Conversely, minimum means lower bound or least possible quantity. The absolute minimum of a function is the smallest number in its range and corresponds to the value of the function at the lowest point of its graph. If f(a) is less than or equal to f(x), for all x in the domain of the function, then f(a) is an absolute minimum. As an example, f(x) = 32x² - 32x - 6 has an absolute minimum of -22, because every value of x produces a value greater than or equal to -22.

In some cases, a function will have no absolute maximum or minimum. For instance the function f(x) = 1/x has no absolute maximum value, nor does f(x) = -1/x have an absolute minimum. In still other cases, functions may have relative (or local) maxima and minima. Relative means relative to local or nearby values of the function. The terms relative maxima and relative minima refer to the largest, or least, value that a function takes on over some small portion or interval of its domain. Thus, if f(b) is greater than or equal to f(b ± h) for small values of h, then f(b) is a local maximum; if f(b) is less than or equal to f(b ± h), then f(b) is a relative minimum. For example, the function f(x) = x⁴ -12x³ - 58x² + 180x + 225 has two relative minima (points A and C), one of which is also the absolute minimum (point C) of the function. It also has a relative maximum (point B), but no absolute maximum.

Finding the maxima and minima, both absolute and relative, of various functions represents an important class of problems solvable by use of differential calculus. The theory behind finding maximum and minimum values of a function is based on the fact that the derivative of a function is equal to the slope of the tangent. When the values of a function increase as the value of the independent variable increases, the lines that are tangent to the graph of the function have positive slope, and the function is said to be increasing. Conversely, when the values of the function decrease with increasing values of the independent variable, the tangent lines have negative slope, and the function is said to be decreasing. Precisely at the point where the function changes from increasing to decreasing or from decreasing to increasing, the tangent line is horizontal (has slope 0), and the derivative is zero. (With reference to figure 1, the function is decreasing to the left of point A, as well as between points B and C, and increasing between points A and B and to the right of point C). In order to find maximum and minimum points, first find the values of the independent variable for which the derivative of the function is zero, then substitute them in the original function to obtain the corresponding maximum or minimum values of the function. Second, inspect the behavior of the derivative to the left and right of each point. If the derivative Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. is negative on the left and positive on the right, the point is a minimum. If the derivative is positive on the left and negative on the right, the point is a maximum. Equivalently, find the second derivative at each value of the independent variable that corresponds to a maximum or minimum; if the second derivative is positive, the point is a minimum, if the second derivative is negative the point is a maximum.

A wide variety of problems can be solved by finding maximum or minimum values of functions. For example, suppose it is desired to maximize the area of a rectangle inscribed in a semicircle. The area of the rectangle is given by A = 2xy. The semicircle is given by x² + y² = r², for y ≥ 0, where r is the radius. To simplify the mathematics, note that A and A² are both maximum for the same values of x and y, which occurs when the corner of the rectangle intersects the semicircle, that is, when y² = r² - x². Thus, we must find a maximum value of the function A² = 4x²(r² -x²) = 4r²x² - 4x⁴. The required condition is that the derivative be equal to zero, that is, d(A²)/dx = 8r²x - 16x³ = 0. This occurs when x = 0 or when x = ¹⁄₂(r √ +2 ). Clearly the area is a maximum when x = ¹⁄₂(r √ +2 ). Substitution of this value into the equation of the semicircle gives y = ¹⁄₂(r √ +2 ), that is, y = x. Thus, the maximum area of a rectangle inscribed in a semicircle is A = 2xy = r².

There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and engineering. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business it is usually possible to express profit as a function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In other cases, the shape of a container may be determined by minimizing the amount of material required to manufacture it. The design of piping systems is often based on minimizing pressure drop which in turn minimizes required pump sizes and reduces cost. The shapes of steel beams are based on maximizing strength.

Finding maxima or minima also has important applications in linear algebra and game theory. For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The quantity to be maximized (or minimized), as well as each of the constraints, is represented by an equation or inequality. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system. A practical example might be minimizing the cost of producing an automobile given certain known constraints on the cost of each part, and the time spent by each laborer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.