Using Derivatives to Solve Optimization Problems

In many applied problems, we're looking for an optimal point, where the error is lowest, for example, or the profit is highest. The traditional way is to model the situation using a function, find the derivative of the function, and solve for the input that makes the derivative zero. This is because the derivative is zero at local minima and maxima, as shown in the following figure:

Figure 10.21: A cubic function and the points we want to find

The function we're given in the figure is f(x) = x3 - 2.8x2 + 1.2x + 0.85. We're interested in finding the local maximum, point A, and the local minimum, point B. We would have to differentiate the function and solve the resulting equation by hand. But using a computer, we can simply start at a value of x on the left of the grid and take small steps, checking f(x) until we get a change in direction. To do that, we can use our derivative function to check...