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# Lesson 18: Maximum and Minimum Values (handout)

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There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.

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### Lesson 18: Maximum and Minimum Values (handout)

1. 1. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Sec on 4.1 Maximum and Minimum Values V63.0121.001: Calculus I Professor Ma hew Leingang New York University April 4, 2011 . . Notes Announcements Quiz 4 on Sec ons 3.3, 3.4, 3.5, and 3.7 next week (April 14/15) Quiz 5 on Sec ons 4.1–4.4 April 28/29 Final Exam Monday May 12, 2:00–3:50pm . . Notes Objectives Understand and be able to explain the statement of the Extreme Value Theorem. Understand and be able to explain the statement of Fermat’s Theorem. Use the Closed Interval Method to ﬁnd the extreme values of a func on deﬁned on a closed interval. . . . 1.
2. 2. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Outline Introduc on The Extreme Value Theorem Fermat’s Theorem (not the last one) Tangent: Fermat’s Last Theorem The Closed Interval Method Examples . . Notes Optimize . . Notes Why go to the extremes? Ra onally speaking, it is advantageous to ﬁnd the extreme values of a func on (maximize proﬁt, minimize costs, etc.) Many laws of science are derived from minimizing principles. Maupertuis’ principle: “Ac on is minimized through the wisdom Pierre-Louis Maupertuis of God.” (1698–1759) . . . 2.
3. 3. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Design . . Notes Optics . . Notes Outline Introduc on The Extreme Value Theorem Fermat’s Theorem (not the last one) Tangent: Fermat’s Last Theorem The Closed Interval Method Examples . . . 3.
4. 4. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Extreme points and values Deﬁni on Let f have domain D. The func on f has an absolute maximum (or global maximum) (respec vely, absolute minimum) at c if f(c) ≥ f(x) (respec vely, f(c) ≤ f(x)) for all x in D The number f(c) is called the maximum value (respec vely, minimum value) of f on D. An extremum is either a maximum or a . minimum. An extreme value is either a maximum value or minimum value. . Image credit: Patrick Q . Notes The Extreme Value Theorem Theorem (The Extreme Value Theorem) maximum value Let f be a func on which is f(c) con nuous on the closed interval [a, b]. Then f a ains minimum an absolute maximum value value f(c) and an absolute minimum f(d) value f(d) at numbers c and d . a d c in [a, b]. b minimum maximum . . Notes No proof of EVT forthcoming This theorem is very hard to prove without using technical facts about con nuous func ons and closed intervals. But we can show the importance of each of the hypotheses. . . . 4.
5. 5. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Bad Example #1 Example Consider the func on { x 0≤x<1 . f(x) = | x − 2 1 ≤ x ≤ 2. 1 Then although values of f(x) get arbitrarily close to 1 and never bigger than 1, 1 is not the maximum value of f on [0, 1] because it is never achieved. This does not violate EVT because f is not con nuous. . . Notes Bad Example #2 Example Consider the func on f(x) = x restricted to the interval [0, 1). There is s ll no maximum value (values get arbitrarily close to 1 but do not achieve it). This does not violate EVT . | because the domain is 1 not closed. . . Notes Final Bad Example Example 1 The func on f(x) = is con nuous on the closed interval [1, ∞). x . 1 There is no minimum value (values get arbitrarily close to 0 but do not achieve it). This does not violate EVT because the domain is not bounded. . . . 5.
6. 6. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Outline Introduc on The Extreme Value Theorem Fermat’s Theorem (not the last one) Tangent: Fermat’s Last Theorem The Closed Interval Method Examples . . Notes Local extrema Deﬁni on A func on f has a local maximum or rela ve maximum at c if f(c) ≥ f(x) when x is near c. This means that f(c) ≥ f(x) for all x in some open interval containing c. |. | local local b a Similarly, f has a local minimum maximum minimum at c if f(c) ≤ f(x) when x is near c. . . Notes Local extrema So a local extremum must be inside the domain of f (not on the end). A global extremum that is inside the domain is a local extremum. |. | a b local local and global maximum global max min . . . 6.
7. 7. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Fermat’s Theorem Theorem (Fermat’s Theorem) Suppose f has a local extremum at c and f is diﬀeren able at c. Then f′ (c) = 0. |. | a local local b maximum minimum . . Notes Proof of Fermat’s Theorem Suppose that f has a local maximum at c. If x is slightly greater than c, f(x) ≤ f(c). This means f(x) − f(c) f(x) − f(c) ≤ 0 =⇒ lim ≤0 x−c x→c+ x−c The same will be true on the other end: if x is slightly less than c, f(x) ≤ f(c). This means f(x) − f(c) f(x) − f(c) ≥ 0 =⇒ lim ≥0 x−c x→c− x−c f(x) − f(c) Since the limit f′ (c) = lim exists, it must be 0. . x→c x−c . Notes Meet the Mathematician: Pierre de Fermat 1601–1665 Lawyer and number theorist Proved many theorems, didn’t quite prove his last one . . . 7.
8. 8. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Outline Introduc on The Extreme Value Theorem Fermat’s Theorem (not the last one) Tangent: Fermat’s Last Theorem The Closed Interval Method Examples . . Flowchart for placing extrema Notes Thanks to Fermat Suppose f is a c is a . start con nuous local max func on on the closed, bounded Is c an no Is f diﬀ’ble no f is not interval endpoint? at c? diﬀ at c [a, b], and c is a global yes yes maximum c = a or ′ point. f (c) = 0 c = b . . Notes The Closed Interval Method This means to ﬁnd the maximum value of f on [a, b], we need to: Evaluate f at the endpoints a and b Evaluate f at the cri cal points or cri cal numbers x where either f′ (x) = 0 or f is not diﬀeren able at x. The points with the largest func on value are the global maximum points The points with the smallest or most nega ve func on value are the global minimum points. . . . 8.
9. 9. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Outline Introduc on The Extreme Value Theorem Fermat’s Theorem (not the last one) Tangent: Fermat’s Last Theorem The Closed Interval Method Examples . . Notes Extreme values of a linear function Example Find the extreme values of f(x) = 2x − 5 on [−1, 2]. Solu on So Since f′ (x) = 2, which is never zero, we have no cri cal points The absolute minimum and we need only inves gate (point) is at −1; the the endpoints: minimum value is −7. f(−1) = 2(−1) − 5 = −7 The absolute maximum (point) is at 2; the f(2) = 2(2) − 5 = −1 maximum value is −1. . . Extreme values of a quadratic Notes function Example Find the extreme values of f(x) = x2 − 1 on [−1, 2]. Solu on We have f′ (x) = 2x, which is zero when x = 0. So our points to check are: f(−1) = 0 f(0) = − 1 (absolute min) f(2) = 3 (absolute max) . . . 9.
10. 10. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Extreme values of a cubic function Example Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2]. Solu on . . Notes Extreme values of an algebraic function Example Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2]. Solu on . . Notes Extreme values of another algebraic function Example √ Find the extreme values of f(x) = 4 − x2 on [−2, 1]. Solu on . . . 10.
11. 11. . V63.0121.001: Calculus I . Sec on 4.1: Max/Min .Values April 4, 2011 Notes Summary The Extreme Value Theorem: a con nuous func on on a closed interval must achieve its max and min Fermat’s Theorem: local extrema are cri cal points The Closed Interval Method: an algorithm for ﬁnding global extrema . . Notes . . Notes . . . 11.