Calculus I Notes, Section 4-2

Notes, Lesson 4.2
Maximum and Minimum Values

Optimal Design

We have learned several derivative techniques. We know that derivatives give us the slope of a curve at any instant (where the function is differentiable). If we then look for times when the slope is zero, we will know when the curve reaches at least a local maximum or local minimum. To be able to find maxima and minima is an extremely powerful ability. This lesson explores this ability. We will be able to solve optimization problems. We can actually find optimal ways of doing things. Just imagine the power of a businessman who can find how to minimuze costs and maximize profit! Imagine the importance of maximizing health quality! Imagine the wonderful by-products of minmizing our damage to the environment! Imagine the safety, speed, efficiency of fuel consuption when optimizing is applied to studying flight efficiency! Shortly (after lesson 4.6) your instructor will show you the presentations "Optimal Design," and "Thinking God's Thoughts after Him," both from the series: "Divine Design" which will explore many of these issues.

Definitions

Absolute Maximum (Global Maximum)	The maximum value of the function over its entire domain.
Absolute Minimum (Global Minimum)	The minumum value of the function over its entire domain.
Local Maximum (Relative Maximum)	The maximum value of the function near a specified value in the domain.
Local Minimum (Relative Minimum)	The minimum value of the function near a specified value in the domain.
Extreme Values	The absolute maximum and absolute minumum of a function.
Extreme Value Theorem	If f is continuous over a closed interval, then it must attain an absolute maximum and an absolute minimum in that interval.
Fermat's Theorem	If a function as a local maximum or local minimum at c, and if f '(c) exists, then f '(c)=0.
Critical Number	A critical number of a function f is a number c in the domain of f such that either f'(c)=0 or f'(c) does not exist.
Closed Interval Method	To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1. Find the values of f at the critical numbers of f in (a,b) 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; and the smallest of these values is the aboslute minimum value.

Example Problem:

Find the absolute maximum and absolute minimum values of on the interval [-2,4].	Given example problem.
	We find the derivative so that we can find critical points.
	Here are our two critical numbers.
	We are now following step 1 from the Closed Interval Method as we find the values of f at the critical numbers.
	We now follow step 2 from the Closed Interval Method as we find the values of f at the endpoints of the interval.
Absolute Minimum = -12 Absolute Maximum = 48	We pick the largest and smallest from steps 1 and 2.

Absolute Extrema and Optimization

Check Concepts

#1: True or False: A critical number of a function occurs whenever the function equals zero.

#2: True or False. To find an absolute minimum on a closed interval, you should check the values of the function at the endpoints of the interval.

#3: True or False: If a function is not differentiable at a point (x), then it can't have a local or global minimum or maximum at that point.

#4: True or False: If f'(x)=0 then there must be a local maximum or minimum at f(x).

#5 True or False: Finding maxima and minima are important in business, manufacturing, scientific research, medicine, and many other fields of study.