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**4.1 Maximum and Minimum Values**

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**Absolute Maximum and Minimum**

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.

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**Local Maximum and Minimum**

Definition: A function f has a local maximum (or relative maximum) at c if f(c) ≥ f(x) when x is near c. Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c.

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Example: Absolute maximum (also local maximum) Local maximum Local minimum Local minimum Absolute minimum (also local minimum)

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**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

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Example when a continuous function doesn’t have minimum or maximum because the interval is not closed. No Maximum No Minimum

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

Absolute maximum (also local maximum) Local maximum Local minimum Notice that local extremes in the interior of the function occur where is zero or is undefined.

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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).

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**Example when f ′(c) = 0 but f has no maximum or minimum at c .**

(not an extreme)

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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) = x1/2(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 .

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**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]: Find the values of f at the critical numbers of f in (a,b) . Find the values of f at the endpoints of the interval. 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.

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