1 / 19

Section 4.1 – Extreme Values of Functions

Section 4.1 – Extreme Values of Functions. Using the first derivative to find maximum and minimum values (max. and min.) Find critical points to locate these values. Absolute (Global) Extreme Values. Definition – Absolute Extreme Values Let f be a function with domain D. Then f(c) is the

jeanne
Télécharger la présentation

Section 4.1 – Extreme Values of Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 4.1 – Extreme Values of Functions • Using the first derivative to find maximum and minimum values (max. and min.) • Find critical points to locate these values

  2. Absolute (Global) Extreme Values • Definition – Absolute Extreme Values Let f be a function with domain D. Then f(c) is the (a) absolute maximum value on D if and only if f(x) ≤ f(c) for all x in D. (b) absolute minimum value on D if and only if f(x) ≥ f(c) for all x in D.

  3. Theorem 1 – 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.

  4. Example 1 (y = x2) • y = x2 , (- , ) • No absolute maximum • Absolute minimum of 0 at x = 0

  5. Example 2 (y = x2) • y = x2 , [0,2] • Absolute maximum of 4 at x = 2 • Absolute minimumof 0 at x = 0

  6. Example 3 (y = x2) • y = x2 , (0,2] • Absolute maximumof 4 at x = 2 • No absolute minimum

  7. Example 4 (y = x2) • y = x2 , (0,2) • No absolute extrema

  8. Possible locations…

  9. Possible locations…(2)

  10. Absolute vs. Local

  11. Local Extreme Values • Definition – Local Extreme Values Let c be an interior point of the domain of the function f. Then f(c) is a (a) local maximum value at c if and only iff(x) ≤ f(c) for all x in some open interval containing c. (b) local minimum value at c if and only iff(x) ≥ f(c) for all x in some open interval containing c.

  12. Theorem 2 – 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 f’ exists at c, then f’(c) = 0 • Definition – Critical PointA point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a critical point of f.

  13. Bellringer Find the derivative of x3– x2 – 3x + 1

  14. Example 1 (Graphically) f(x) = x3– x2 – 3x + 1 f’(x) = x2 – 2x – 3 D: [-4, 6]

  15. Example 1 (Analytically) f(x) = x3 – x2– 3x + 1 D: [-4, 6] f’(x) = x2 – 2x – 3 x2 – 2x – 3 = 0 f(-4) = -24.333 (x - 3)(x + 1) = 0 f(-1) = 2.667 x = -1, 3 f(3) = -8 f(6) = 19

  16. Example 2 f(x) = What is our domain? Why? f’(x) = We already know our function is undefined at certain points Critical point is at x = 0 f(0) =

  17. Steps to check • Is the function undefined or 0 at any points? • Find the derivative and find any places the derivative is not defined or is equal to 0. • Use these values in f(x) and find their values.Check the domain values in f(x) as well. • Determine if they are max. or min. values(Global or Local)

  18. Example 3 f(x) = ln | | Undefined? f‘(x) = Critical points? Check f(x) values? Maximums or minimums?

  19. Further Examples • Pg. 184 (1, 3, 5, 19, 21)

More Related