1. MAT 213 Brief Calculus Section 4.2 Relative and Absolute Extreme Points

2. Let’s see what we remember about derivatives of a function and its graph –If f’ > 0 on an interval than f is Increasing –If f’ < 0 on an interval than f is Decreasing –If f’(a) = 0 than f is Neither increasing nor decreasing Has a horizontal tangent at a

3. Use the derivative to determine where each function is increasing and where it is decreasing. Verify by looking at the graphs. f(x)=x 2 – 4x + 7 g(x)=x 3 - 3x 2

4. In the previous example, f(x)=x 2 - 4x + 7, the point x=2 is called a critical point because f’(2) = 0

5. Critical Points For any function f, a point p in the domain of f where f’(p) = 0 or f’(p) is undefined is called a critical point of the function –The critical value of f is the function value, f(p) where p is the critical point –Critical points are used to determine relative extrema

6. Relative Extrema f has a local (relative) maximum at x = p if f(p) is equal to or larger than all other f values near p –If p is a critical point and f’ changes from positive to negative at p, then f has a local maximum at p f has a local (relative) minimum at x = p if f(p) is equal to or smaller than all other f values near p –If p is a critical point and f’ changes from negative to positive at p, then f has a local maximum at p Since in the previous cases we were using the first derivative, we were using the first derivative test to check for relative extrema

7. The First Derivative Test for Local Maxima and Minima Suppose p is a critical point of a continuous function f. If f’ changes from negative to positive at p, then f has a local minimum at p. If f’ changes from positive to negative at p, then f has a local maximum at p.

8. Often we want to find the maximum or minimum value of a function This is called optimization Absolute (Global) Maxima and Minima –f has an absolute (or global) minimum at p if f(p) is less than or equal to all values of f –f has a global maximum at p if f(p) is greater than or equal to all values of f Absolute (or global) maxima and minima are sometimes referred to as “extrema” or “optimal values”

9. We may not always have a relative extrema and we may not always have absolute extrema Let’s take a look at the following graphs and identify –Relative and absolute maxima –Relative and absolute minima –Concavity –Inflection points

15. The number of Nonbusiness Chapter 11 Bankruptcies can be modeled by a cubic polynomial The cubic would have the following equation with the plot on the next slide Years (Since1998) Bankruptcies 0981 1731 2722 3745 4894 5966 6935 Consider the following situation B(t) = −12.58t 3 + 134.1t 2 − 358.8t + 979.2

16. Where are the relative extrema in this graph? Where are the absolute extrema in this graph?

17. Now in the previous graph we had relative extrema, but no absolute extrema Recall that our model was from 1998 (or t = 0) to 2004 (or t = 6) Thus we must have absolute extrema between those years Let’s take a look at the graph from that point of view and see if we can identify any absolute extrema

19. Extreme Value Theorem If f is a continuous function on the closed interval a ≤ x ≤ b, then f has a global maximum and a global minimum on that interval If f is continuous over a closed interval, we are guaranteed to have an absolute max and absolute min They either occur at critical points or at the endpoints Thus the procedure is to find all critical points of f, evaluate f at the critical points and endpoints and compare values

20. To find the absolute extrema of a continuous function on a closed interval (i.e. endpoints are included): –Compare function values at the critical points and endpoints To find the absolute of a continuous function on an open interval (i.e. endpoints not included or infinite endpoint): –Find the value of the function at all critical points and sketch the graph. –Look at the function values when x approaches the endpoints of the interval, or approaches ±∞, when appropriate. Let’s try 27 from the book