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3.1 Extrema On An Interval

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**After this lesson, you should be able to:**

Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval

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Definition

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Extrema Minimum and maximum values on an interval are called extremes, or extrema on an interval. The minimum value of the function on an interval is considered the absolute minimum on the interval. The maximum value of the function on an interval is considered the absolute maximum on the interval. When the just word minimum or maximum is used, we assume it’s an absolute min or absolute max.

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**OPEN intervals – Do the following have extrema?**

On an open interval, the max. or the min. may or may not exist even if the function is continuous on this interval.

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**CLOSED intervals – Do the following have extrema?**

On a closed interval, both max. and min. exist if the function is continuous on this interval.

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**The Extreme Value Theorem (EVT)**

Theorem 3.1: If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval. In other words, if f is continuous on a closed interval, f must have a min and a max value. Max-Min f is continuous on [a, b] a b

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Example Example 1 Let f (x) = x2 – 5x – 6 on the closed interval [–1, 6], find the extreme values.

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Example Example 2 Let f (x) = x3 + 2x2 – x – 2 on the closed interval [–3, 1], find the extreme values.

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Example Example 3 Let f (x) = x3 + 2x2 – x – 2 on the closed interval [–3, 2], find the extreme values. The (absolute)max and (absolute)min of f on [a, b] occur either at an endpoint of [a, b] or at a point in (a, b).

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**Relative Extrema and Critical Numbers**

(AP may use Local Extrema)

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If there is an open interval containing c on which f (c) is a maximum, then f (c) is a local maximum of f. If there is an open interval containing c on which f (c) is a minimum, then f (c) is a local minimum of f. When you look at the entire graph (domain), there may be no absolute extrema, but there could be many relative extrema. What is the slope at each extreme value????

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**Definition of a Critical Number and Figure 3.4**

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**Critical Numbers c is a critical number for f iff:**

f (c) is defined (c is in the domain of f ) f ’(c) = 0 or f ’(c) = does not exist Theorem If f has a relative max. or relative min, at x = c, then c must be a critical number for f. The (absolute)max and (absolute)min of f on [a, b] occur either at an endpoint of [a, b] or at a critical number in (a, b). So…. Relative extrema can only occur at critical values, but not all critical values are extrema. Explain this statement.

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****Make sure you give the y-value this is the extreme value!****

Guidelines Make sure f is continuous on [a, b]. Find the critical numbers of f(x) in (a, b). This is where the derivative = 0 or is undefined. Evaluate f(x) at each critical numbers in (a, b). Evaluate f(x) at each endpoint in [a, b]. The least of these values (outputs) is the minimum. The greatest is the maximum. **Make sure you give the y-value this is the extreme value!**

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**Critical Numbers To find the max and min of f on [a, b]:**

Make sure f is continuous on [a, b]. Find all critical numbers c1, c2, c3…cn of f which are in (a, b) where f’(x) = 0 or f’(x) is undefined. Evaluate f(a), f(b), f(c1), f(c2), …f(cn). The largest and smallest values in part 2 are the max and min of f on [a, b].

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**Example (–, +) x = 2 and x = 5 Example 4 Find all critical numbers**

Domain: (–, +) Critical number: x = 2 and x = 5

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**Example Example 5 Find all critical numbers. Domain: x ≠ 1, xR**

x = 1, x = 0, and x = 2

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**Example (–, +) Example 6 Find all critical numbers. Domain: f’(–4 )**

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**Example (–, +) x = –2 and x = 4**

Example 7 Find the max and min of f on the interval [–3, 5]. Domain: (–, +) Critical number: x = –2 and x = 4 Graph is not in scale x Left Endpoint Critical Number Right Endpoint f (x) f (–3)= 20 f (–2)= 30 f (4)=–78 f (5)=–68 maximum minimum

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**Practice Of (–, +) x = 0 and x = 1**

Example 7 Find the extrema of f on the interval [–1, 2]. Domain: (–, +) Critical number: x = 0 and x = 1 x Left Endpoint Critical Number Right Endpoint f (x) f (–1)= 7 f (0)= 0 f (1)=–1 f (2)=16 minimum maximum

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f ’(0) does not exist Example Example 8 Find the extrema of f on the interval [–1, 3]. Critical number: x = 0 and x = 1 x Left Endpoint Critical Number Right Endpoint f (x) f (–1)= –5 f (0)= 0 f (1)=–1 f (3)= minimum maximum

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**Practice Of x = /2, x = 3/2, x = 7/6, x = 11/6**

Example 8 Find the extrema of f on the interval [0, 2]. Critical number: x = /2, x = 3/2, x = 7/6, x = 11/6 x Left Endpoint Critical Number Right Endpoint f (x) f (0)=–1 f (/2)= 3 f (7/6) =–3/2 f (3/2) =–1 f (11/6) =–3/2 f (2) =–1 max min min

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**Summary Open vs. Closed Intervals 1. An open interval MAY have extrema**

2. A closed interval on a continuous curve will ALWAYS have a minimum and a maximum value The min & max may be the same value How?

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Homework Section 3.1 page 165 #1-23 odd, 55

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