Math 151-copyright Joe Kahlig, 13C
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Sections 5.2 and 5.3: Maximum and Minimum Values, Derivatives and Shapes of Curves
Definition: A critical number (critical value) is a number, c, in the domain of f such that
f ′ (c) = 0 or f ′ (c) DNE.
If f has a local extrema (local maxima or minima) at c then c is a critical value of f (x).
Example: Find the intervals where the function is increasing and the intervals where it is decreasing.
Classify all critical values.
A) y = x3 + 3x2 − 9x + 8
B) y = 3x5 − 20x3 + 20
C) y =
x2 + 1
x
Math 151-copyright Joe Kahlig, 13C
D) y = (x2 − 16)2/3
E) y = x ln(x)
F) y = xex
2 −3x
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Math 151-copyright Joe Kahlig, 13C
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Definition: x = c is a possible inflection value (piv) provided that x = c is in the domain of f (x)
and f ′′ (c) = 0 or f ′′ (c) DNE.
Example: Find the intervals where the function is concave up and the intervals where it is concave
down. Find the x-coordinate of the inflection points.
A) y = x5 − 5x4 + 10x + 5
B) y = x ln(x − 2)
Example: Find the values of a and b so that f (x) = ax2 − b ln(x) will have an inflection point at (1, 5)
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Math 151-copyright Joe Kahlig, 13C
Example: The domain of the function f (x) is all real numbers except x = −5. Use this information
as well as f ′ and f ′′ to sketch a possible graph for f (x).
f ′ (x) =
−3x + 7
(x + 5)3
f ′′ (x) =
6(x − 6)
(x + 5)4
Example: Suppose that f has critical values of x = 0, x = 2, and x = −2. If f ′′ (x) = 60x3 − 120x,
what conclusion can be drawn about the critical values?
Definition: Suppose that f ′′ is continuous near the critical value c.
(a)
(b)
(c)
Example: What conclusion can be made if you know that g′′ (5) = 7?
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Math 151-copyright Joe Kahlig, 13C
Absolute Maxima and Minima
Definition: A function f has an absolute maximum (global maximum) at c if f (c) ≥ f (x) for
all x in D, where D is the domain of f . f (c) is called the maximum value of f on D. Similarly, f has
an absolute minimum at c if f (c) ≤ f (x) for all x in D and the number f (c) is call the minimum
value of f on D. The maximum and minimum values of f are called the extreme values of f .
Example: For these functions, find the absolute max and the absolute min.
A) y = x3 + 3x2 + 1
B) y = x4 − 4x3
Restricted Domains:
J
K
J
K
J
K
Math 151-copyright Joe Kahlig, 13C
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Definition: If f is a continuous function on a closed interval [a, b], then f will have both an absolute
max and an absolute min. They will happen at either critical values in the interval or at the ends of
the interval, x = a or x = b.
Example: For the function, find the absolute max and the absolute min on the indicated interval.
f (x) = 12x2 − 2x3 + 1
A) [2, 5]
B) [−3, 5]
C) (−3, 5]
Example: For the function, find the absolute max and the absolute min on the interval [0, 5].
f (x) =
1
(x − 4)2
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Math 151-copyright Joe Kahlig, 13C
−π π
,
.
Example: For the function, find the absolute max and the absolute min on the interval
2 2
f (x) = cos(x)
The Mean Value Theorem: If f is a differentiable function on the interval [a, b], then there exist
a number c between a and b such that
f ′ (c) =
f (b) − f (a)
b−a
Example: Find a number c that satisfies the conclusion of the Mean Value Theorem on the interval
[0, 2].
f (x) = x3 + x − 1
Example: You enter a toll road at 8am and then exit it at 9:15am. The distance between the entrance
and exit is 100 miles. If the maximum speed is set at 70mph, do you get charged for speeding?