AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.1: Extreme Values of Functions

What you’ll learn about Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values …and why Finding maximum and minimum values of a function, called optimization, is an important issue in real-world problems.

Absolute Extreme Values Absolute maximum and minimum values are also called absolute extrema. The term “absolute” is often omitted.

The Extreme Value Theorem

The Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval. 2 conditions for f: continuous & closed interval If either condition does not exist the E.V.T. does not apply.

Classifying Extreme Values

Local Extreme Values

Local Extreme Values

Extreme Values can be Absolute or Local Local (relative) Extreme Values Let c be an interior point of the domain of the function f. Then f(c) is a Local maximum value at c if and only if f(x) ≤ f(c) for all x in some open interval containing c. b) Local minimum value at c if and only if f(x) ≥ f(c) for all x in some open interval containing c. A function f may have a local max or local min at an endpoint c if the appropriate inequality holds. An extreme value may be local and global.

Critical Points

Section 4.1 – Extreme Values of Functions Not all critical numbers (points) may be actual relative minimums or maximums. f’(c)=0, but no max or min minimum where f’(c) does not exist

Example Finding Absolute Extrema

Example Finding Extreme Values

Section 4.1 – Extreme Values of Functions Theorem: Local Extreme Values If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, then In other words, if f has a relative maximum or relative minimum at x = c, then c is a critical number of f.

Section 4.1 – Extreme Values of Functions (Absolute) Finding Extrema on a Closed Interval Find the critical numbers of f in Evaluate f at each critical number in Evaluate f at each endpoint of The least of these values is the absolute minimum. The greatest of these values is the absolute maximum.

Graphically Look at the graph. The highest point on the graph is the global / absolute maximum. The lowest point on the graph is the global / absolute minimum. Don’t forget to consider the endpoints when looking at a closed interval! Find the maximum and minimum points of f(x) = cos x on [-π, π]

Example 3 Finding Absolute Extrema Find the absolute maximum and minimum values of f(x) = x 2/3 on the interval [-2,3]. Find x values where f ’ = 0 (make sure they are in the domain) Check endpoints of domain Find the y-values of each critical point to determine the maximum and minimum points. Practice: Find absolute extrema of g(x)=e-x on [-1,1].

Finding Extreme Values Find the extreme values of Derivative = 0 Endpoints Find y-values to evaluate

Make Sure to Answer the Question being Asked! What is the maximum value?  The output value is crucial. Where does the maximum occur?  The input value is crucial. Where on the curve is the maximum?  The POINT (ordered pair) is crucial.

Summary Absolute extrema is on a closed interval and can be at the critical numbers (derivative = 0 or undefined) or at the endpoints of the interval. Relative/local extrema are the maximum/minimum values that the function takes on over smaller open intervals. There are no endpoints to test like we do for absolute extrema.

You try: What are the extreme values of

You try: Find the absolute maximum and minimum values of

Section 4.1 – Extreme Values of Functions Finding the Relative (local) maximum or minimum of a function (using a graphing calculator) Enter the equation in y=. Graph in the appropriate window. Use (2nd) CALC, maximum or minimum (On the TI-89, use F5 (Math), max or min). Arrow to just left of the critical point, ENTER. Arrow to just right of the critical point, ENTER. Guess? – Just press ENTER

Section 4.1 – Extreme Values of Functions To find approximations of the roots of a polynomial using a graphing calculator: Enter the equation in y=. Graph in the appropriate window. Use (2nd) CALC, ZERO (on the TI-89, use F% (Math) ZERO). Arrow to just left of the zero, ENTER. Arrow to just right of the zero, ENTER. GUESS? – press ENTER.