1. Section 4.1Maximum and Minimum Values AP Calculus October 20, 2009 Berkley High School, D1B1 todd1@toddfadoir.com

2. Max and Min • Of course they can’t be too easy; this IS calculus. • Absolute Maximum: f has an absolute maximum if there exists a c such that f(c)≥f(x), for x in D. f(c) is called the maximum value. • Absolute Minimum: f has an absolute minimum if there exists a c such that f(c)≤f(x), for x in D. f(c) is called the minimum value. • The absolute max and absolute min are called the extremes. Calculus, Section 4.1

3. Max and Min • Local Maximum: f has a local maximum if there exists a c such that f(c)≥f(x), for all x near c. • How close to c? Calculus close. • Local Minimum: f has a local minimum if there exists a c such that f(c)≤f(x), for all x near c. • Endpoints of closed intervals can not be local maximums or minimums. Calculus, Section 4.1

4. Example Local Minimum and Absolute Minimum Local Maximum Local Maximum and Absolute Maximum Local Minimum Calculus, Section 4.1

5. Local Minimums Local Minimums Local Minimums Example Local Minimum Local and Absolute Maximum Local Maximum Absolute Minimum Calculus, Section 4.1

6. Example Local Maximum Local and Absolute Maximum Absolute Minimum Local Minimum Local Minimum Calculus, Section 4.1

7. Critical Numbers • Definition: “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.” • Theorem: “If f has a local maximum or minimum at c, then c is a critical number of f.” • Translation: Any critical number has the potential of being a local maximum of minimum. Only at critical numbers can a local max or a local min exist. Calculus, Section 4.1

8. Example x=0 is a critical value, but not a local maximum or minimum Calculus, Section 4.1

9. Example f(0)=0 is a local minimum. Because the derivative at 0 is undefined, 0 is a critical value. Calculus, Section 4.1

10. Closed Interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: • Find the values of f at the critical numbers of f on (a,b) • Find the values of f at the endpoints of the interval. • The largest of the values for steps 1 & 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Calculus, Section 4.1

11. Example Calculus, Section 4.1

12. Example Absolute Minimum Absolute Maximum Calculus, Section 4.1

13. Assignment • Section 4.1, 15-55, odd Calculus, Section 4.1