Section 5.3: Extrema, Inflection Points, and Graphing
Theorem: (First Derivative Test)
Suppose that c is a critical number of a continuous function f .
1. If f 0 changes from positive to negative at c, then f has a local maximum at c.
2. If f 0 changes from negative to positive at c, then f has a local minimum at c.
3. If f 0 does not change sign at c, then f has no local maximum or minimum at c.
Example: Use the First Derivative Test to identify all local extrema.
(a) f (x) = x3 − 3x + 1
(b) f (x) =
x
x2 + 4
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Theorem: (Second Derivative Test)
Suppose that c is a critical number of f and f 00 is continuous on some interval containing c.
1. If f 00 (c) > 0, then f has a local minimum at c.
2. If f 00 (c) < 0, then f has a local maximum at c.
3. If f 00 (c) = 0, the test is inconclusive.
Example: Use the Second Derivative Test to identify all local extrema.
(a) f (x) = 2x3 + 3x2 − 12x + 11
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(b) f (x) = x4 − 2x3 − 6x2 + 2
2
2
Example: Let f (x) = 2x3 + 3x2 − 36x.
(a) Determine where f is increasing or decreasing.
(b) Find the local extrema of f .
(c) Determine where f is concave up or down and identify the inflection points.
(d) Sketch the graph of f (x).
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Example: Let f (x) = x4 − 2x2 + 3.
(a) Determine where f is increasing or decreasing.
(b) Find the local extrema of f .
(c) Determine where f is concave up or down and identify the inflection points.
(d) Sketch the graph of f (x).
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Definition: The line x = a is a vertical asymptote of y = f (x) if either
lim f (x) = ±∞
x→a−
or
lim f (x) = ±∞.
x→a+
The line y = L is a horizontal asymptote of y = f (x) if either
lim f (x) = L
x→−∞
or
lim f (x) = L.
x→∞
Example: Find all vertical and horizontal asymptotes of f (x) =
5
x2 + 3x
.
x2 + 8x + 15
Definition: The line y = mx + b is an oblique asymptote of y = f (x) if either
lim [f (x) − (mx + b)] = 0
x→−∞
or
lim [f (x) − (mx + b)] = 0.
x→∞
These types of asymptotes occur with a rational function in which the degree of the numerator
is one higher than the degree of the denominator.
Example: Find all asymptotes of f (x) =
x3 − 2x2
.
x2 − 5x − 6
6
x2
.
x2 − 1
(a) Find all vertical and horizontal asymptotes of f (x).
Example: Let f (x) =
(b) Determine where f is increasing or decreasing and identify the local extrema.
(c) Determine where f is concave up or down and identify the inflection points.
(d) Sketch the graph of f (x).
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Example: Let f (x) =
x2
.
x+1
(a) Find all asymptotes of f (x).
(b) Determine where f is increasing or decreasing and identify the local extrema.
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(c) Determine where f is concave up or down and identify the inflection points.
(d) Sketch the graph of f (x).
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