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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

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Section 4.1 – Extreme Values of Functions

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  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)

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