MATH 12002 - CALCULUS I § 3.1: Maximum and Minimum Values Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 10
Absolute Maximum & Minimum Definition Let y = f ( x ) be a function and let a be a number in the domain of f . We say that f has an absolute maximum at x = a if f ( a ) � f ( x ) for all x in the domain of f . The y value f ( a ) is the absolute maximum value of f . We say that f has an absolute minimum at x = a if f ( a ) � f ( x ) for all x in the domain of f . The y value f ( a ) is the absolute minimum value of f . Thus the absolute maximum value of a function is just the largest y value and the absolute minimum value is the smallest y value of the function. D.L. White (Kent State University) 2 / 10
Absolute Maximum & Minimum A function may or may not have an absolute maximum or minimum value. 1 f ( x ) = x 2 with domain [0 , 1] has absolute minimum y = 0 at x = 0 and absolute maximum y = 1 at x = 1. r ✻ 1 r ✲ 1 2 f ( x ) = x 2 with domain [0 , 1) has absolute minimum y = 0 at x = 0 ❜ but no absolute maximum. ✻ 1 r ✲ 1 3 f ( x ) = x 2 with domain (0 , 1] has absolute maximum y = 1 at x = 1 r but no absolute minimum. ✻ 1 ❜ ✲ 1 D.L. White (Kent State University) 3 / 10
Absolute Maximum & Minimum 4 f ( x ) = x 3 with domain ( −∞ , ∞ ) has neither an absolute minimum ❛ nor an absolute maximum. q ✻ q � x + 1 3 � ❅ q if 0 � x < 2 2 � ❅ 5 f ( x ) = 1 ✲ 4 − x if 2 � x � 4 1 2 3 4 has absolute minimum y = 0 at x = 4 but has no absolute maximum. Observe: Functions (2)–(4) are continuous but are not defined on closed intervals. Function (5) is defined on the closed interval [0 , 4] but is not continuous. Function (1) is defined on the closed interval [0 , 1] and is continuous. D.L. White (Kent State University) 4 / 10
Absolute Maximum & Minimum In fact, the only conditions that can prevent a function from having both an absolute minimum and an absolute maximum are discontinuity or a domain that is not a closed interval. Extreme Value Theorem If y = f ( x ) is a continuous function on a closed interval [ a , b ] , then f has both an absolute maximum and an absolute minimum on [ a , b ] . D.L. White (Kent State University) 5 / 10
Local Maximum & Minimum Definition Let y = f ( x ) be a function and let c be a number in the domain of f . We say that f has a local maximum at x = c if f ( c ) � f ( x ) for all x in some open interval around x = c . We say that f has a local minimum at x = c if f ( c ) � f ( x ) for all x in some open interval around x = c . Thus f has a local maximum at x = c if the y values for x near c are all less than or equal to f ( c ). Similarly, f has a local minimum at x = c if the y values for x near c are all greater than or equal to f ( c ). D.L. White (Kent State University) 6 / 10
Local Maximum & Minimum These occur at the “peaks” and “valleys” of the graph. q q ✻ f ✛ ✲ a c e d b q q q ❄ The function f has: a local maximum at x = c and at x = e , and a local minimum at x = d . Note also: f has an absolute minimum at x = a , and an absolute maximum at x = c . However, f does not have a local minimum at x = a , because f is not defined in an open interval around x = a . D.L. White (Kent State University) 7 / 10
Local Maximum & Minimum Notes: In order to have a local maximum or minimum at x = c , f must be defined in some open interval around c . Hence if c is an endpoint of the domain of f , there can be no local maximum or minimum at x = c . If f has an absolute maximum or minimum at x = c and c is not an endpoint of the domain of f , then f has a local maximum or minimum at x = c as well. From the Extreme Value Theorem and this note, we can deduce Theorem If y = f ( x ) is a continuous function on a closed interval [ a , b ] , then f has an absolute maximum and an absolute minimum on [ a , b ] and these must occur at one of the endpoints (x = a or x = b) or at a point c where a < c < b and f has a local maximum or minimum. D.L. White (Kent State University) 8 / 10
Local Maximum & Minimum We have reduced the problem of how to determine the absolute maximum and minimum values of a continuous function on a closed interval [ a , b ], to the problem of how to determine the local maximum and minimum values on the open interval ( a , b ). For this, we use the derivative. The derivative may not exist at a local maximum or minimum, as in ❅ � ❅ � ✪ ✫ or ❅ � However, if the derivative does exist, then there is a tangent line, ✬ ✩ q and the tangent line must be horizontal, as in ✫ ✪ q or Hence at a local maximum or minimum, either the derivative does not exist, or it must be 0. D.L. White (Kent State University) 9 / 10
Local Maximum & Minimum Theorem If y = f ( x ) has a local maximum or minimum at x = c, then either f ′ ( c ) = 0 or f ′ ( c ) is undefined. Notes: A number c in the domain of f where f ′ ( c ) = 0 or f ′ ( c ) is undefined is called a critical number for f . We can have f ′ ( c ) = 0 or f ′ ( c ) undefined without having a local maximum or minimum at x = c . √ x at x = 0.) (For example, f ( x ) = x 3 or f ( x ) = 3 This theorem tells us that the only place we have to look for a local maximum or minimum is where f ′ ( c ) = 0 or f ′ ( c ) is undefined. We conclude that a continuous function y = f ( x ) defined on [ a , b ] has an absolute maximum and an absolute minimum, and these occur either at an endpoint ( x = a or x = b ) or at some critical number c with a < c < b . D.L. White (Kent State University) 10 / 10
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