1. Extrema On An Interval Section 3.1

2. After this lesson, you should be able to: • Understand the definition of extrema of a function on an interval • Understand the definition of relative extrema of a function on an open interval • Find extrema on a closed interval

3. Extrema Minimum and maximum values on an interval are called extremes, or extrema on an interval. • The minimum value of the function on an interval is considered the absolute minimum on the interval. • The maximum value of the function on an interval is considered the absolute maximum on the interval.

4. f is continuous on [a, b] f a b The Extreme Value Theorem Theorem 3.1: If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval.

5. Examples • Given , name any extrema of f on the interval • [0, 5] • (0, 5)

6. e.g. In these cases, the graph has a horizontal tangent line at the relative max/min; i.e. e.g. Relative Extrema • Relative extrema are turning points of the graph. • The turning points may occur as smooth “hills” or “valleys”. • The turning points may occur as sharp turns. In these cases, the function is not differentiable at the relative max/min.

7. Critical Numbers • c is a critical number for f iff: • f(c) is defined (c is in the domain of f) • f ´(c) = 0 or f ´(c) does not exist • If f has a relative max. or relative min, at x = c, then c must be a critical number for f. • The (absolute)max and (absolute)min of f on [a, b] occur either at an endpoint of [a, b] or at a critical number in (a, b).

8. Finding Extrema on a Closed Interval To find the max and min of f on [a, b]: • Find all critical #s of f which are in (a, b). Find all values of x for which f ´(c) = 0 or f ´(c) does not exist • Evaluate f at each of the critical values Plug each of the critical values into the function to find the y-coordinate. • Evaluate f at each endpoint Find f(a) and f(b) • The smallest value from parts 2 & 3 is the minimum and the largest value from parts 2 & 3 is the maximum of f on [a, b].

9. Example Example: Find all critical numbers Domain:

10. Example Example: Find all critical numbers. Domain:

11. Example Example: Find all critical numbers. Domain:

12. x f(x) Example Example: Find the max and min of f on the interval [0, 4]. Domain:

13. x f(x) Example Example: Find the max and min of f on the interval [-1, 1]. Domain:

14. Example Example: Graph a function f on the interval [-3, 4] that has the given characteristics. • Relative max at x = -2 • Absolute min at x = 2 • Absolute max at x = 4

15. Homework Section 3.1 page 169 #1-25 odd, 37, 45