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Extreme values of functions

Extreme values of functions. Section 4.1a. Definition: Absolute Extreme Values. Let be a function with domain D . Then is the. absolute maximum value on D if and only if for all x in D. (b) absolute minimum value on D if and only if for all x in D.

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Extreme values of functions

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  1. Extreme values of functions Section 4.1a

  2. Definition: Absolute Extreme Values Let be a function with domain D. Then is the • absolute maximum value on D if and only if • for all x in D. (b) absolute minimum value on D if and only if for all x in D. Absolute (or global) maximum and minimum values are also called absolute extrema. Sometimes the word “absolute” or “global” is simply omitted… The Extreme Value Theorem (EVT) If is continuous on a closed interval [a, b], then has both a maximum value and a minimum value on the interval.

  3. For our first example, let’s complete the following table: Domain D Absolute Extrema on D Function No absolute max. Absolute min. of 0 at x = 0 Abs. max. of 4 at x = 2 Abs. min at (0, 0) Abs. max. of 4 at x = 2 No absolute min. No abs. extrema How about a graph for each of these???

  4. Definition: Local (Relative) Extreme Values Let c be an interior point of the domain of the function . Then is a • local maximum value at c if and only if • for all x in some open interval containing c. (b) local minimum value at c if and only if for all x in some open interval containing c. Note: all absolute extrema are also local extrema!!!

  5. Abs. Max. Local Max. Local Min. Abs. Min. Local Min. x Let’s classify extreme values in the function shown above

  6. Guided Practice – Identify Any Extreme Values y 7 3 x –7 –4 –2 5 10 Local Min of 0 at x = – 7 –6 Max of 7 at x = – 4 and x = 5 Min of – 6 at x = – 2 Local Min of 3 at x = 10

  7. The “Do Now” Writing: How can the derivative help in identifying extreme values of functions (maxima and minima)? Theorem: Local Extreme Values If a function has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, then

  8. Definition: Critical Point A point in the interior of the domain of a function at which or does not exist is a critical point of . • To find extremaanalytically, we only • need to investigate the critical points • and the endpoints of the function!!!

  9. Guided Practice Find the extreme values of on the interval [–2, 3]. First, find the first derivative: Next, identify any critical points: No zeros, but undefined at x = 0 (i.e., the function has a critical point at x = 0)

  10. Guided Practice Find the extreme values of on the interval [–2, 3]. Finally, evaluate and analyze the original function at this critical point, and at the endpoints:  Abs. Min. Crit. Pt: Endpts:  Local Max.  Abs. Max. Can we support these answers graphically???

  11. Guided Practice Find the extreme values of  Find answer both analytically and graphically! Domain is all real numbers…need to check critical points… The derivative is never undefined. The derivative is zero for x = 1. As x moves away from 1 in either direction, f increases. Absolute Minimum of 3 at x = 1

  12. Guided Practice Find the extreme values of the given function. The derivative has no zeros. We only need to check endpoints. Maximum value is e at x = –1 Minimum value is 1/e at x = 1

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