4.7 Optimization Problems

In this section we solve such problems as maximizing areas, volumes, and pro ts and minimizing distances, times, and costs.

What does optimization mean?

Optimization is the process of finding the maximum or minimum value of a function using derivatives. 

We use derivatives because maxima and minima occur at critical points.

Steps in Solving Optimization Problems

1. Understand the problem

• Identify what quantity is being maximized or minimized

2. Draw a diagram

• Visualizing the situation often makes relationships clearer

3. Define the Variables

• Choose symbols for the unknown quantities.

4. Write the primary equation

• Form an equation for the quantity to be maximized or minimized.

5. Write the Secondary Constraint

• Use given information to relate the variables.

• Then use these equations to eliminate all but one of the variables in the

  expression for Q.

6. Use Methods from 4.1 and 4.3 to find the absolute max and min value of f  

• In particular, if the domain of f is a closed interval, then the Closed

  Interval Method in Section 4.1 can be used.

First Derivative Test for absolute Extreme Values 

The first derivative test is used to find local extrema, while the process for finding absolute extrema on a closed interval involves evaluating the function at its critical points and the endpoints of the interval.

First Find the critical points and evaluate the original function at these critical points  and the interval’s endpoints. 

The largest and smallest values among these results are the absolute maximum and minimum

Finding Optimization using Implicit Differentiation

To find an optimization using implicit differentiation, find the first derivative of the objective function (treating y as a function of x) by differentiating both sides of the equation, and then set the derivative equal to zero to find critical points.

Finally, use the second derivative test to determine if the critical points are maxima or minima. 

Summary

Optimization applies derivatives to find maxima and minima of real-world quantities.

We build a function, reduce it to one variable, differentiate to find critical points, and test them.