Maxima and Minima
Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using Differential Calculus. Consider the graph shown below.
As x increases, the curve rises if the slope is positive, as of arc AB; it falls if the slope is negative, as of arc BC.
Relative Maximum and Minimum Points
At a point such as B, where the function is algebraically greater than that of any neighboring point, the point is said to have a maximum value, and the point is called a maximum point (relative to adjacent points). Similarly at D, the function has a minimum value (relative to adjacent points). At maximum or minimum points, the tangent is horizontal or the slope is zero.
dy/dx = y' = 0
This does not necessarily mean that at these points the function is maximum or minimum. It does only mean that the tangent is parallel to the x-axis, or the curve is either concave up or concave down. The points at which dy/dx = 0 are called critical points, and the corresponding values of x are critical values.
The second derivative of a function is the rate of change of the first derivative or the rate of change of the slope. It follows that as x increases and y” is positive, y’ is increasing and the tangent turns in a counterclockwise direction and the curve is concave upward. When y” is negative, y’ decreases and the tangent turns in the clockwise direction and the curve is concave downward.
If y’ = 0 and y” is negative (i.e. y” < 0), the point is a maximum point (concave downward).
If y’ = 0 and y” is positive (i.e. y” > 0), the point is a minimum point (concave upward).
Points of Inflection
A point of inflection is a point at which the curve changes from concave upward to concave downward or vice versa (see point E from the figure). At these points the tangent changes its rotation from clockwise to counterclockwise or vice versa.
At points of inflection, the second derivative of y is zero (y” = 0).
Application of Maxima and Minima
As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as A = f(x). The common task here is to find the value of x that will give a maximum value of A. To find this value, we set dA/dx = 0.
Steps in Solving Maxima and Minima Problems
- Identify the constant, say cost of fencing.
- Identify the variable to be maximized or minimized, say area A.
- Express this variable in terms of the other relevant variable(s), say A = f(x, y).
- If the function shall consist of more than one variable, expressed it in terms of one variable (if possible and practical) using the conditions in the problem, say A = f(x).
- Differentiate and equate to zero, dA/dx = 0.
