section 3.1 – extrema on an interval

Section 3.1 – Extrema on an Interval

Maximum Popcorn Challenge You wanted to make an open-topped box out of a rectangular sheet of

paper 8.5 in. by 11 in. The student must cut congruent squares out of each corner of the sheet and then bend the edges of the sheet upward to form the sides of the box. For what dimensions does the box have the greatest possible volume?

xx

11

Draw a Picture

What needs to be Optimized?Volume needs to be maximized:

V lwh

Eliminate Variable(s) with other Conditions

11 2 8.5 2V x x x 3 24 39 93.5V x x x

Use Calculus to Solve the Problem2' 12 78 93.5V x x 20 12 78 93.5x x

278 78 4 12 93.52 12x

11 2 1.585 7.829l 8.5 2 1.585 5.329w 1.585 in x 7.829 in

x 5.329 in

xx

8.5

8.5

– 2

x

11 – 2x

1.585,4.915x You can’t cut an

4.9x4.9 in. square out of an 8.5x11 in. paper

x varies from box to

box

The slope of a tangent is 0 at a max

Quad. Form.

A Beginning to Optimization Problems

One of the principal goals of calculus is to investigate the behavior of various functions. There exists a large class of problems that involve finding a maximum or minimum value of a function, if one exists. These problems are referred to as optimization problems and require an introduction to terminology and techniques.

Example of an optimization problem:A manufacturer wants to design an open box having a

square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

Let f be a function defined on an interval I that contains the number c. Then:

These values are also referred to as maximum/minimum, extreme values, or absolute extrema.

Extrema of a Function

f(c) is an absolute maximum of f on I if f(c) ≥ f(x) for all x in I.

f(c) is an absolute minimum of f on I if f(c) ≤ f(x) for all x in I.

c

c

f(c)

f(c)

II

Example 1The graph of a function f is shown below. Locate the

extreme values of f defined on the closed interval [a,b].

a c d e f b

f(x)

x

The highest point occurs at x=b

The lowest point occurs at x=d

Absolute Maximum: Absolute Minimum:f(b) f(d)


extreme values of f defined on the open interval (0,1).

0.5

1

f(x)

x

The function may have a limit at the highest point BUT

there is no absolute

maximum value

Absolute Maximum: Absolute Minimum:None None

x y

.9 .9

.99 .99.999 .999

.9999 .9999

… …

The function may have a limit at the lowest point BUT

there is no absolute minimum

value

x y

.1 .1

.01 .01.001 .001

.0001 .0001

… …


extreme values of f defined on the closed interval [-1,1].

1

0.5

f(x)

x

Absolute Maximum: Absolute Minimum:2 None

-0.5

The highest point occurs at x=1 & -1

There is no lowest point because a discontinuity

exists at the border

There is an issue because this function is not continuous on the closed interval [-1,1]

White Board ChallengeSketch a graph of the function with the

following characteristics:

It is defined on the open interval (-7,-1).

It is not differentiable at x=-4

It has a maximum of 5 and a minimum of -4.

The Extreme Value TheoremA function f has an absolute maximum and an absolute

minimum on any closed, bounded interval [a,b] where it is continuous.

a c d e f b

f(x)

x

Absolute Maximum

Absolute Minimum


Key Word.

This function is continuous and defined

on the intervals.

Example 1In each case, explain why the given function does not

contradict the Extreme Value Theorem:

1

1 2

f(x)

x

Even though the function has no maximum, it does not contradict

the EVT because it is no continuous on [0,2].

Even though the function has no minimum, it does not contradict

the EVT because it is not defined on a closed interval.

a. f x 2x if 0 x 11 if 1x 2

b. g x x 2 on 0 x 2

2

1 2

g(x)

x

White Board ChallengeThe function below describes the position a

particle is moving in a horizontal straight line.

Find the average velocity between t = 2 and 4.

2100 20 5x t t

10vt

A function f has a relative maximum (or local maximum) at c if f(c) ≥ f(x) when x is near c. [This means that f(c) ≥ f(x) for all x in some open interval containing c.]

A function f has a relative minimum (or local minimum) at c if f(c) ≤ f(x) when x is near c. [This means that f(c) ≤ f(x) for all x in some open interval containing c.]

Relative Extrema of a Function

a c d e f b

f(x)

x

Typically relative extrema

of continuous functions occur at “peaks” and

“valleys.”

f(c) is a relative

maximum at x=c

f(d) is a relative

minimum at x=d

f(e) is a relative

maximum at x=e

f(f) is a relative minimum at

x=f

Endpoints are not relative

extrema.

Plural = Relative maxima/minima

Relative Extrema and DerivativesSince relative extrema exist at “peaks” and

“valleys,” this suggests that they occur when:

The derivative is zero (horizontal tangent)

The derivative does not exist (no tangent)

Critical Numbers and Critical PointsSuppose f is defined at c and either f '(c)=0 or f '(c)

does not exist. Then the number c is called a critical number of f, and the point (c, f(c)) on the graph of f is called a critical point.

-3 is a critical number and (-3,7) is a critical point

2 is a critical number and (2,3) is a critical point

Example 1Find the critical numbers for . 3 24 5 8 20f x x x x

Domain of Function:All Real Numbers

3 2' 4 5 8 20ddxf x x x x

Take the Derivative

3 2' 4 5 8 20d d d ddx dx dx dxf x x x x

2' 12 10 8 0f x x x

Now find when the derivative is 0 and/or undefined for x

values in the domain.

20 12 10 8x x

Solve the Derivative for 0

20 2 6 5 4x x

0 2 3 4 2 1x x 4 13 2,x

The derivative is defined for all real numbers.

2' 12 10 8f x x x Both values are in

the domain.When is the derivative undefined?

Example 2Find the critical numbers for . 2

2x

xf x

Domain of Function: All Real Numbers except 2

2

2' d xdx xf x

Take the Derivative

2 2

2

2 2

2'

d ddx dxx x x x

xf x

2

2

2 2 1

2' x x x

xf x



20 4x x


0 4x x

0, 4x

The derivative is not defined for x = 2.

2 2

22 4

2' x x x

xf x

Both values are in the domain.

2

242

' x xx

f x

BUT x = 2 is not in the domain of the function.

When is the derivative undefined?

White Board ChallengeConsider the function below:

Find the equation of the tangent line to the function at the vertex.

24 6 19f x x x

21.25y

Example 3Find the critical numbers for . 2 6f x x x

Domain of Function:All Real Numbers greater than or equal to 0

1 2' 2 6ddxf x x x

Take the Derivative

1 2 3 2' 12 2ddxf x x x

1 2 3 2' 12 2d ddx dxf x x x



1 2 1 20 6 3x x


1 2 1 22x x2x

The derivative is not defined for 0 or negative numbers.

and 0

1 2 1 2' 6 3f x x x

2 is in the domain.


Since 0 is in the domain, it is also a critical point.

Example 4Find the critical points for . 21 2f x x x


2' 1 2ddxf x x x Take the Derivative

2 2' 1 2 2 1d ddx dxf x x x x x

2' 1 1 2 2 1f x x x x

Now find when the derivative is 0 and/or undefined for x values in the domain.

0 1 3 3x x Solve the Derivative for 0

1, 1x

The derivative is defined for all real numbers.

' 1 1 2 2f x x x x

Both values are in the domain.


' 1 3 3f x x x

Find the y-value(s) 21 1 1 1 2f

21 1 1 1 2f

4

0 1,4 1,0and

Example 5Find the critical numbers for . 1f x x


1 11 1

x xf x

x x

Take the Derivative

1 1'

1 1x

f xx




1x

The derivative is undefined for x=-1.

Since -1 is in the domain


The derivative never equals 0.

Critical Number Theorem

If a continuous function has a relative extremum at c, then c must be a critical number of f.

NOTE: The converse is not necessarily true. In other words, if c is a critical number of a continuous function f, c is NOT always a relative extremum.

Important NoteNot every critical point is a relative extrema.

3y x2' 3y x

20 3x0x

30 0y

0,0

Take the Derivative


Find the y-value(s)

is NOT a relative extrema

White Board ChallengeFind the derivative of the function below:

sin secf x x

' cos sec sec tanf x x x x

How do we Find Absolute Extrema?Suppose we are looking for the absolute extrema of a

continuous function f on the closed, bounded interval [a,b]. Since the EVT says they must exist, how can we narrow the list of candidates for points where extrema exist?

a c d e f b

f(x)

x

Absolute Maximum

Absolute Minimum


On a closed interval, extrema exist at endpoints

or at relative extrema.

Procedure for Finding Absolute Extrema on an Closed Interval

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.

Summary of ProcedureFind the absolute maximum and minimum of the function

graphed below.The value of the function at the

critical number 2 is:

-3

Find the values of f at critical numbers

The value of the function at the enpoint 0 is:

1

Find the values of f at the endpointsThe value of the function at the enpoint 3 is:

-2Find the largest and smallest values from the above work

smallest

largest

1 is the maximum and -3 is the minimum

Example 1Find the absolute extrema of the function defined by the

equation on the closed interval [-1,2]. 4 22 3f x x x

4 2' 2 3ddxf x x x


3' 4 4f x x x 30 4 4x x

The maximum occurs at x=2 and is 11; the minimum

occurs at x=-1 and 1 and is 2

20 4 1x x

Not a critical point

since it’s an

enpoint

Answer the Question 0 4 1 1x x x

0,1, 1x

4 21 1 2 1 3f 2

4 20 0 2 0 3f 3

Find the values of f at the endpoints

4 22 2 2 2 3f 11

4 21 1 2 1 3f 2smallest

largest

smallest

Domain of f: All Reals


equation on the closed interval [0,2π]. 2sinf x x x

' 2sinddxf x x x


' 1 2cosf x x

0 1 2cos x

The maximum occurs at x=5π/3 and is 6.97; the

minimum occurs at x= π/3 and is -0.68

2cos 1x

Answer the Question

12cos x

5 5 53 3 32sinf 6.968039

3 3 32sinf 0.684853


2 2 2sin 2f 6.28

0 0 2sin 0f 0

largest

smallest

53 3,x



equation on the closed interval [-1,2]. 2 3 5 2f x x x

2 3 5 3' 5 2ddxf x x x


1 3 2 310 103 3'f x x x

1 3 2 310 103 30 x x

The maximum occurs at x=-1 and is 7; the minimum occurs

at x=0 and is 0

1 31030 1x x

Answer the Question

1x

2 30 0 5 2 0f 0

2 31 1 5 2 1f 3


2 32 2 5 2 2f 1.5874

2 31 1 5 2 1f 7largest

smallest

x=0 is a critical number too since it makes the derivative undefined.


section 3.1 – extrema on an interval

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