MATH 12002 - CALCULUS I 3.1: Maximum and Minimum Values Professor - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §3.1: Maximum and Minimum Values

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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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) = x2 with domain [0, 1] has absolute minimum y = 0 at x = 0

and absolute maximum y = 1 at x = 1.

✲ ✻

1 1

r r

2 f (x) = x2 with domain [0, 1) has absolute minimum y = 0 at x = 0

but no absolute maximum.

✲ ✻

1 1

r ❜

3 f (x) = x2 with domain (0, 1] has absolute maximum y = 1 at x = 1

but no absolute minimum.

✲ ✻

1 1

❜ r

D.L. White (Kent State University) 3 / 10

Absolute Maximum & Minimum

4 f (x) = x3 with domain (−∞, ∞) has neither an absolute minimum

nor an absolute maximum.

5 f (x) =

x + 1 if 0 x < 2 4 − x if 2 x 4

✲ ✻

1 2 3 1 2 3 4

q

• ❛

q ❅ ❅q

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].

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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.

✲ ✛ ✻ ❄

a c d e b

f q q 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.

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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)

• r 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

• f 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

✪ ✫

• r
• ❅

❅ ❅

However, if the derivative does exist, then there is a tangent line, and the tangent line must be horizontal, as in

✫ ✪ q

• r

✬ ✩ q

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. (For example, f (x) = x3 or f (x) =

3

√x at x = 0.) 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)

• r at some critical number c with a < c < b.

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