MTH 251 – Differential Calculus Chapter 4 – Applications of Derivatives Section 4.1 Extreme Values of Functions Copyright © 2010 by Ron Wallace, all rights reserved.

Terminology The following are the same …  Absolute (aka: Global) Extrema  Absolute (aka: Global) Minimum and/or Maximum  Extreme Values Basic problem of this chapter …  Determine the extreme values of a function over an interval.  i.e. Given f(x) where x  [a,b] or (a,b) or [a,b) or (a,b] ; determine the largest and/or smallest value of f(x). Note: The extreme values are values of the function. The extreme values occur at one or more values of x in the interval.

Max & Min – Formal Definitions f(x) has an Absolute Maximum over a domain D at a point x = c if and only if f(x) ≤ f(c) for all x in D. f(x) has an Absolute Minimum over a domain D at a point x = c if and only if f(x) ≥ f(c) for all x in D. Note: Absolute Extrema may occur at more than one value of x.

Possible Locations of Extrema Top of a peak Bottom of a valley End point Point of discontinuity  the function must be defined

Do Extrema Exist? Possibilities …  Both max & min?  Max but no min?  Min but no max?  No max or min? The Extreme Value Theorem  If f(x) is continuous over (aka: on) [a,b], then f(x) has a absolute maximum value M and an absolute minimum value m over the interval. Note that m ≤ f(x) ≤ M for all x  [a,b] and … … there exists x 1 & x 2  [a,b] where f(x 1 ) = m and f(x 2 ) = M

Local Extrema If there is some open interval that contains x = c where f(c) is an extrema over that interval, then f(c) is a Local Extrema.  aka: Relative Extrema The left endpoint of the domain of a function is a local extrema. A right endpoint of the domain of a function is a local extrema.

Finding Extrema Some facts …  Absolute extrema are also relative extrema.  Possible locations of relative extrema are the same as absolute extrema i.e. peaks, valleys, endpoints, discontinuities  Peaks & Valleys occur at “critical points” Points where f ’(x) is zero or undefined Note: Not all critical points are extrema

Proof regarding Critical Points If f(c) is a local maximum and f’(c) exists, then f’(c) = 0.  Local Max implies that f(x) ≤ f(c) for some interval containing c. That is, f(x) – f(c) ≤ 0 Since these must be equal … The proof for local minimums would be essentially the same (all of the inequalities would be reversed).

Finding Extrema Some facts …  Absolute extrema are also relative extrema.  Possible locations of relative extrema are the same as absolute extrema i.e. peaks, valleys, endpoints, discontinuities  Peaks & Valleys occur at “critical points” Points where f ’(x) is zero or undefined Note: Not all critical points are extrema Method … for closed intervals 1.Find the values of x of all critical points. i.e. f’(x) = 0 or DNE 2.Calculate f(x) for all critical points and endpoints. 3.The extrema are the largest and smallest of the values in step 2.

Finding Extrema – Example Method … for closed intervals 1.Find the values of x of all critical points. i.e. f’(x) = 0 or DNE 2.Calculate f(x) for all critical points and endpoints. 3.The extrema are the largest and smallest of the values in step 2. Determine the extrema for … 20 3

Extrema on Open Intervals Instead of calculating the value of the function at the endpoint, you must calculate the limit as x approaches the endpoint. Method … for open intervals 1.Find the values of x of all critical points. i.e. f’(x) = 0 or DNE 2.Calculate f(x) for all critical points. 3.Calculate the limits at the endpoints. one sided limits 4.The extrema are the largest and smallest of the values in step 2 provided that they are larger or smaller than the limits in step 3. Note: Semi-open intervals will use a combination of the two previous cases.

Finding Extrema – Example Domain? Method … for open intervals 1.Find the values of x of all critical points. i.e. f’(x) = 0 or DNE 2.Calculate f(x) for all critical points. 3.Calculate the limits at the endpoints. one sided limits 4.The extrema are the largest and smallest of the values in step 2 provided that they are larger or smaller than the limits in step 3. Determine the extrema for …

Determine the extrema for … (note: semi-open) Finding Extrema – Example

Determine the extrema for … (note: open … domain?) Finding Extrema – Example