1. Section 3.1 Calculus AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us
Extrema on an Interval
Section 3.1
Calculus AP/Dual, Revised ©2013
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3.1 - Extrema on an Interval

2. 3.1 - Extrema on an Interval
Pop Quiz
Take a sheet of paper and answer these 8 questions. You have 7 minutes. You do not have to write the question.
sin (π/6) 5) cot (7π/6)
cos (2π/3) 6) csc (7π/4)
tan (–11π/6) 7) sec (π/2)
cot (2π) 8) sin (3π)
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3.1 - Extrema on an Interval

3. 3.1 - Extrema on an Interval
Definitions
A. Relative Extrema vs Absolute Extrema
Relative (Local) - occur on an open interval
Absolute (Global) - occur on a closed interval
If there is an open interval containing c on which is a minimum, then (c, f(c)) is called a relative minimum of f, or you can say that f has a relative minimum at (c, f(c)). It is also called the local minimum. Relative Minimums are like valleys of the graphs.
If there is an open interval containing c on which is a maximum, then (c, f(c)) is called a relative maximum of f, or you can say that f has a relative maximum at (c, f(c)). It is also called the local maximum. Relative Maximums are like hills of the graphs.
There can be infinitely many relative maximums and relative minimums on an interval, but there is only one non-repeating y-value for the absolute max and absolute min.
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3.1 - Extrema on an Interval

4. 3.1 - Extrema on an Interval
Extreme Value Theorem
Extreme Value Theorem is if f is continuous on a closed interval [a, b], then f has both a minimum f (c) < f (x) and a maximum f (c) > f (x) on the interval.
f has to be continuous
Holes do not qualify for maximums or minimums
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5. 3.1 - Extrema on an Interval
Definition of Extrema
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6. 3.1 - Extrema on an Interval
Example 1
Where would the absolute and relative maxima be located on the graph, based on the interval [–2, 4]?
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3.1 - Extrema on an Interval

7. Determine Absolute Extrema & Critical Numbers
To Determine the Absolute Extrema of a Function given the equation:
Identify the Critical Numbers
Check the Endpoints
If f '(c) = 0 (where top equals zero) or if f '(c) = DNE (where the bottom of the derivative equals zero) then c is a critical number of f.
Check where the derivative equals Zero (Peak or Valley on Graph)
Results in horizontal tangent lines
Therefore, a max or min and where the function is not differentiable
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3.1 - Extrema on an Interval

8. Example 2
Establish the Critical Value and any absolute and relative extrema 2 1 2 3 4 5 6
f(x) x 1
Relative Max or Min can involve endpoints IFF the interval is OPEN
Absolute Max or Min can involve endpoints IFF the interval is CLOSED. –1 –2
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9. 3.1 - Extrema on an Interval
Example 3
Identify any Critical Numbers of f(x) = x2(x2 – 4)
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10. 3.1 - Extrema on an Interval
Example 3
Identify any Critical Numbers of f(x) = x2(x2 – 4)
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11. 3.1 - Extrema on an Interval
Example 4
Identify any Critical Numbers of 𝑓 𝑥 = 4𝑥 𝑥 2 −3
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12. 3.1 - Extrema on an Interval
Example 4
Identify any Critical Numbers of 𝑓 𝑥 = 4𝑥 𝑥 2 −3
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13. 3.1 - Extrema on an Interval
Example 5
Identify any Critical Numbers of 𝑓 𝑥 = 𝑥 1/3
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14. 3.1 - Extrema on an Interval
Your Turn
Identify any Critical Numbers of f(x) = (9 – x2)3/5
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15. 3.1 - Extrema on an Interval
Candidates Test
For closed intervals only, identify the critical numbers where f ’(x) = 0 or f ’(x) = DNE
Make a t-chart with x and f(x)
X-candidates are the endpoints and Critical Numbers
Find the associated y-Numbers and identify the max or min to the ORIGINAL equation
Reminder
If the question is asking WHAT is the max or min value in the interval, LOOK AT THE f(x) or Y-Value
If the question is asking WHERE is the max or min value in the interval, LOOK AT THE X-Value
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16. 3.1 - Extrema on an Interval
Example 6
Identify the absolute maxima of f (x) = x2 + 2x – 4 on [–1, 1] x
f (x)
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17. 3.1 - Extrema on an Interval
Example 7
Identify the absolute maxima of x3 – 12x in [0, 4] x
f (x)
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18. 3.1 - Extrema on an Interval
Example 7
Identify the absolute maxima of x3 – 12x in [0, 4]
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19. 3.1 - Extrema on an Interval
Example 8
Find the absolute extrema of f(x) = 3x2/3 – 2x at [–1, 1].
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20. 3.1 - Extrema on an Interval
Example 8
Find the absolute extrema of f(x) = 3x2/3 – 2x at [–1, 1]. x
f (x)
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21. 3.1 - Extrema on an Interval
Your Turn
Find the absolute extrema of 𝑓 𝑡 = 𝑡 𝑡−2 at [1, 5].
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22. 3.1 - Extrema on an Interval
Example 9
Find the absolute extrema of f(x) = sin x + cos x at [0, 2π].
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23. 3.1 - Extrema on an Interval
Your Turn
Find the absolute extrema of f(x) = cos2x + sin x at [0, 2π].
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3.1 - Extrema on an Interval

24. AP Multiple Choice Practice Question (non-calculator)
Determine from the graph whether f possesses extrema on the interval [a, b) [A] Maximum at x = a, minimum at x = b [B] Maximum at x = a, minimum at x = b [C] No extrema [D] No maximum, minimum at x = a [E] None of these
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3.1 - Extrema on an Interval

25. AP Multiple Choice Practice Question (non-calculator)
Find all of the critical numbers for the function, 𝑓 𝑥 = 𝑥−1 𝑥+3 [A] x = 1 [B] x = +1 [C] x = –3 [D] x = –3, x = 1 [E] None of these
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26. 3.1 - Extrema on an Interval
Assignment
Page 169
9-18 all, odd, 33, 39A, 39C
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3.1 - Extrema on an Interval