Maximum and Minimum Values - Calculus I - Lecture Slides, Slides of Calculus

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4.1 Maximum and Minimum

Values

Definition:

• A function f has an absolute maximum (or global maximum ) at c if f(c) ≥ f(x) for all x in D, where D is the domain of f. The number f(c) is called maximum value of f on D.
• Similarly, f has an absolute minimum at c if f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D.
• The maximum and minimum values of f are called the extreme values of f.

Absolute Maximum and Minimum

Absolute minimum (also local minimum)

Local maximum

Local minimum

Absolute maximum (also local maximum)

Local minimum

Example:

Extreme Value Theorem:

If f is continuous over a closed interval, then f has absolute maximum and minimum over that interval.

Maximum & minimum at interior points

Maximum & minimum at endpoints

Maximum at interior point, minimum at endpoint

Local maximum

Local minimum

Notice that local extremes in the interior of the function occur where f^ ′^ is zero or f^ ′ is undefined.

Absolute maximum (also local maximum)

Suppose we know that extreme values exist. How to find them?

Fermat’s Theorem

If f has a local maximum or minimum at

c , and if f ′(c) exists, then f ′(c) = 0.

Note: When f ′(c) = 0 , f doesn’t necessarily have a

maximum or minimum at c. (In other words, the converse of Fermat’s Theorem is false in general).

Critical numbers

• Definition: A critical number of a function f is a number c in the domain of f such that either f ′(c) = 0 or f ′(c) doesn’t exist_._
• Example: Find the critical numbers of f(x) = x 1/2(x-3)

Solution :

Thus, the critical numbers are 0 and 1.

• If f has a local maximum or minimum at c , then c is a critical number of f.

x

x

x

x x x x

x f x 2

3 3

2

3 2

2

3 ( )

−

− +

• =

− ′ =

The closed interval method for finding

absolute maximum or minimum

To find the absolute maximum and minimum values

of a continuous function f on a closed interval

[a,b]:

1. Find the values of f at the critical numbers of f in

(a,b).

2. Find the values of f at the endpoints of the

interval.

3. The largest of the values from Steps 1 and 2 is

the absolute maximum value; the smallest of

these values is the absolute minimum value.

Examples on the board.