3.1: Extrema on an Interval

Extrema of a Function

Definition of Extrema

Let 𝑓 be defined on an interval I containing c.

1. 𝑓(c) is the minimum of 𝑓 on I when 𝑓(c) ≤ 𝑓(x) for all x in I.
2. 𝑓(c) is the maximum of 𝑓 on I when 𝑓(c) ≤ 𝑓(x) for all x in I.

The miminum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum, also called global minimum/maximum, on the interval.

Extrema that occur at the endpoints are called endpoint extrema.

The Extreme Value Theorem

If 𝑓 is continuous on a closed interval [a, b], then 𝑓 has both a minimum and a maximum on the interval.

Definition of Relative Extrema

1. If there is an open interval containing c on which 𝑓(c) is a maximum, then 𝑓(c) is called a relative maximum of 𝑓, or you can say that 𝑓 has a relative maximum at (c,𝑓(c).
2. 2. If there is an open interval containing c on which 𝑓(c) is a maximum, then 𝑓(c) is called a relative minimum of 𝑓, or you can say that 𝑓 has a relative minumum at (c,𝑓(c).

The plural of relative maximum is relative maxima, and the plural of relative minimum is relative minima. Relative maximum and relative minimum are sometimes called local maximum and local minimum, respectively.

Definition of a Critical Number

Let 𝑓 be defined at c. If 𝑓'(c) = 0 or if 𝑓 is not differentiable at c, then c is a critical number of 𝑓.

Relative Extrema Occur Only at Critical Numbers

If 𝑓 has a relative minimum or relative maximum at x = c, then c is a critical number of 𝑓.

Finding Extrema on a Closed Interval

Guidelines for Finding Extrema on a Closed Interval

To find the extrema of a continuous function 𝑓 on a closed interval [a, b], use these steps.

1. Find the critical numbers of 𝑓 in (a, b).
2. Evaluate 𝑓 at each critical number in (a, b).
3. Evaluate 𝑓 at each en point of [a, b].
4. The least of these values is the minimum. the greatest is the maximum.

3.2: Rolle's Theorem and the Mean Value Theorem

Rolle's Theorem

Let 𝑓 be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If 𝑓(a) = 𝑓(b), then there is at least one number c in (a, b) such that 𝑓'(c) = 0.

its so bad that it didn't even get a cool name...

Mean Value Theorem

If 𝑓 is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that

$$f'(c)=\frac{f(b)-f(a)}{b-a}$$

3.3 Increasing and Decreasing Functions and the First Derivative Test

Increasing and Decreasing Functions

Definitions of Increasing and Decreasing Functions

A function 𝑓 is increasing on an interval when, for any two numbers \(x_1\) and \(x_2\) in the interval, \(x_1\) < \(x_2\) implies

\(f(x_1)\) < \(f(x_2)\)

A function 𝑓 is decreasing on an interval when, for any two numbers \(x_1\) and \(x_2\) in the interval, \(x_1\) > \(x_2\) implies

\(f(x_1)\) > \(f(x_2)\)

Test for Increasing and Decreasing Functions

Let 𝑓 be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

1. If 𝑓'(x) > 0 for all x in (a, b), then 𝑓 is increasing on [a, b].
2. If 𝑓'(x) < 0 for all x in (a, b), then 𝑓 is decreasing on [a, b].
3. If 𝑓'(x) = 0 for all x in (a, b), then 𝑓 is constant on [a, b].

Guidelines for Finding Intervals on which a Function is Decreasing

let 𝑓 be continuous on the interval (a, b). To find the open intervals on which 𝑓 is increasing or decreasing, use the following steps.

1. Locate the critical numbers of 𝑓 in (a, b), and use these numbers to determine the test intervals.
2. Determine the sign of 𝑓'(x) at one test value in each of the intervals.
3. Use the Test for Increasing and Decreasing Functions to determine whether 𝑓 is increasing or decreasing on each interval.

The First Derivative Test

The First Derivative Test

Let c be a critical number of a function 𝑓 that is continuous on an open interval I containing c. If 𝑓 is differentiable on the interval, except possibly at c, then 𝑓(c) can be classified as follows.

1. If 𝑓'(x) changes from negative to positive at c, then 𝑓 has a relative minimum at (c,𝑓(c)).
2. If 𝑓'(x) changes from positive to negative at c, then 𝑓 has a relative maximum at (c,𝑓(c)).
3. If 𝑓'(x) is positive on both sides of c or negative on both sides, then 𝑓(c) is neither a relative minimum nor a relative maximum.

3.4: Concavity and the Second Derivative Test

Concavity

Definition of Concavity

Let 𝑓 be differentiable on an open interval I. The graph of 𝑓 is concave upward on I when 𝑓' is increasing on the interval and concave downward on I when 𝑓' is decreasing on the interval.

Test for Concavity

Let 𝑓 be a function whose second derivative exists on an open interval I.

1. If 𝑓''(x) > 0 for all x in I, then the graph of 𝑓 is concave upward on I.
2. If 𝑓''(x) < 0 for all x in I, then the graph of 𝑓 is concave downward on I.

Points of Inflection

Definition of Point of Inflection

Let 𝑓 be a function that is continuous on an open interval, and let c be a point in the interval. If the graph of 𝑓 has a tangent line at this point (c,𝑓(c)), then this point is a point of inflection of the graph of 𝑓 when the concavity of 𝑓 changes from upward to downward (or vice versa).

Points of Inflection

If (c,𝑓(c)) is a point of inflection of the graph 𝑓, then either 𝑓''(c) = 0 or 𝑓'' does not exist at x = c.

The Second Derivative Test

Second Derivative Test

Let 𝑓 be a function such that 𝑓' (c) = 0 and the second derivative of 𝑓 exists in an open interval containing c.

1. If 𝑓'' > 0, then 𝑓 has a relative minimum at (c,𝑓(c)).
2. If 𝑓'' < 0, then 𝑓 has a relative maximum at (c,𝑓(c)).

If 𝑓''(c) = 0, then the test fails. That is, ~~you suck at math and should retire~~ 𝑓 may have a relative maximum, relative minimum, or neither.

In such cases, you can use the First Derivative Test.

3.5: A Summary of Curve Sketching

Analyzing the Graph of a Function

This is all we've leartn about analyzing graphs so far:

• x-intercepts and y-intercepts.
• Symmetry.
• Donain and Range.
• Continuity.
• Vertical Asymptotes.
• Differentiability.
• Relative extrema.
• Concavity.
• Points of inflection.
• Horizontal Asymptotes.
• Infinite limites at infinity.

Guindelines for Analyzing the Graph of a Function

1. Determine the domain and range of the function.
2. Determine the intercepts, asymptotes and symmetry of the graph.
3. Locate the x-values for which 𝑓'(x) and 𝑓''(x) are either zero or don't exist. Use the results to determine relative extrema and points of inflection.

Actual Guidelines for Analyzing the Graph of a Function

1. Find absolute maximums and minimums (if possible).
2. Determine where the function is increasing/decreasing.
   1. Find the first derivative.
   2. Determine all Critical Numbers
   3. Make a sign chart
   4. Apply the Test for Increasing and Decreasing Functions
3. Find all relative minimums and maximum.
   1. Using the previous sign chart, apply the First Derivative Test.
   2. 2.
4. willfinishlater talkt ot me about it or smth

3.6: Optimization Problems

Applied Minimum and Maximum Problems

Guidelines for Solving Applied Minimum and Maximum Problems