4.1 Extreme Values of Functions Objective: SWBAT determine the local or global extreme values of a function

Chapter 4 is all about applications of derivatives. One of the most useful things we can learn from a function’s derivative is whether the function assumes any maximum or minimum values on a given interval and where those values are located (if they exist). Finding a functions’ extreme values can help us answer questions such as “what is the most effective size for a dose of medicine?” and “what is the least expensive way to pipe oil from an offshore well to a refinery down the coast?”

Extrema (plural for extremum) are maximum or minimum values of a function. Absolute (global) extrema means the biggest or smallest y-value in an interval. – Often, the term “absolute” or “global” is omitted, and you are just asked to find the maximum or minimum value. Meaning: 1) the y-value at x=c is a minimum if that y-value is smaller than all the other y-values for the entire interval and 2) the y-value at x=c is a maximum if that y-value is larger than all the other y-values for the entire interval

Relative (local) extrema exist when the value of a function is larger (or smaller) than all other function values relatively close to that value.

Note: A function can only have one absolute maximum and one absolute minimum, but can have several relative max and mins. It is fairly straightforward getting these values from a graph. But how do we get them algebraically? Maybe derivatives will help…

Note 1: Extreme values occur only at critical points and endpoints. Note 2: Just because a derivative is equal to zero (or undefined) does not mean there is a relative maximum or minimum there. Relative extrema can ONLY occur at critical points (or endpoints), and critical points occur ONLY when the derivative is either 0 or undefined. However, it is possible for the derivative to equal zero (or be undefined) and there be NO extrema there.

Guidelines for Finding Absolute/Relative Extrema on a Closed Interval 1)Find the critical numbers of f in (a,b). Do this by finding where f’=0 or f’ is undefined. 2)These critical numbers AND the endpoints make up a list of candidates for the extrema. Evaluate each candidate by plugging these numbers into the original function. 3)The least of the values from step 2 is the absolute minimum, and the greatest of these values is the absolute maximum. If the interval is closed and the endpoints do not result in an absolute max or min, a sign chart can be used to determine whether or not the endpoints result in a relative max or min.

Critical numbers (or critical points) are x-values. Maximums/minimums of a function are y-values. – In other words, if the point (2,70) gives us a relative min, then the minimum of the function is 70 and it occurs at x=2. – This is how to correctly describe extrema.

Sign Chart!!!

Sign Chart