section 3.1 – extrema on an interval
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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)
Example 2The graph of a function f is shown below. Locate the
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
… …
Example 3The graph of a function f is shown below. Locate the
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
Absolute Maximum: Absolute Minimum:f(b) f(d)
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
Now find when the derivative is 0 and/or undefined for x
values in the domain.
20 4x x
Solve the Derivative for 0
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
Now find when the derivative is 0 and/or undefined for x
values in the domain.
1 2 1 20 6 3x x
Solve the Derivative for 0
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.
When is the derivative undefined?
Since 0 is in the domain, it is also a critical point.
Example 4Find the critical points for . 21 2f x x x
Domain of Function:All Real Numbers
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.
When is the derivative undefined?
' 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
Domain of Function:All Real Numbers
1 11 1
x xf x
x x
Take the Derivative
1 1'
1 1x
f xx
Now find when the derivative is 0 and/or undefined for x
values in the domain.
Solve the Derivative for 0
1x
The derivative is undefined for x=-1.
Since -1 is in the domain
When is the derivative undefined?
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
Solve the Derivative for 0
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
Absolute Maximum: Absolute Minimum:f(b) f(d)
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
Find the values of f at critical numbers
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
Example 2Find the absolute extrema of the function defined by the
equation on the closed interval [0,2π]. 2sinf x x x
' 2sinddxf x x x
Find the values of f at critical numbers
' 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
Find the values of f at the endpoints
2 2 2sin 2f 6.28
0 0 2sin 0f 0
largest
smallest
53 3,x
Domain of f: All Reals
Example 3Find the absolute extrema of the function defined by the
equation on the closed interval [-1,2]. 2 3 5 2f x x x
2 3 5 3' 5 2ddxf x x x
Find the values of f at critical numbers
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
Find the values of f at the endpoints
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.
Domain of f: All Reals