In using quantitative techniques in business, we often try to maximize profit or minimize cost. If a profit function or cost function can be developed, taking a derivative may help us to find the optimum solution. In dealing with nonlinear functions, we often look at local optimums, which represent the maximums or minimums within a small range of X.

Figure M6.4 illustrates a curve where point A is a local maximum (it is higher than the points around it), and point B is a local minimum (it is lower than the points around it); however, there is no global maximum or minimum, as the curve continues to increase without bound as X increases, and to decrease without bound as X decreases. If we place limits on the maximum and minimum values for X, then the endpoints can be checked to see if they are higher than any local maximum or lower than any local minimum.

To find a local optimum, we take the derivative of the function and set it equal to zero. ­Remember that the derivative gives the slope of the function. For a point to be a local maximum or minimum, the tangent line must be a horizontal line, which has a slope of zero. Therefore, when we set the derivative equal to zero and solve, we find a value of X that might be a local maximum or minimum. Such a point is called a critical point.

The following function generated the graph in Figure M6.4:

Point A gives a local maximum and point B gives a local minimum. To find the values of X where these occur, we find the first derivative and set this equal to zero:

Solving this for X, we factor this and have

so the critical points occur when $X=2$ and $X=6.$

The second derivative is

At the first critical point, $X=2,$ so

Because this is negative, this point is a local maximum. At the second critical point, where $X=8,$

Because this is positive, this critical point is a local minimum.

Consider Figure M6.5, which is a graph of

This derivative is equal to zero when $X=0.$ The second derivative is

When $X=0,Y\mathrm{"}=6\left(0\right)=0.$ Thus, this is neither a maximum nor a minimum but is a point of inflection, as shown in Figure M6.5.

A critical point will be:

1. A maximum if the second derivative is negative.

2. A minimum if the second derivative is positive.

3. A point of inflection if the second derivative is zero.