Section 5.3: Derivatives and the Shapes of Curves
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: Find where each function is increasing or decreasing and identify all local extrema.
(a) f (x) = x3 − 3x + 1
(b) f (x) = e−x
2 /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
(b) f (x) =
x2
x
+4
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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Example: Let f (x) = sin x + cos x, 0 ≤ x ≤ 2π.
(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) = x2 ln x.
(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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x2
.
x2 − 1
(a) Find the vertical and horizontal asymptotes.
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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Theorem: (Mean Value Theorem)
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number
c ∈ (a, b) such that
f (b) − f (a)
f 0 (c) =
.
b−a
Example: Show that f (x) = x3 + x − 1 satisfies the hypotheses of the Mean Value Theorem
on [0, 2] and find all values of c that satisfy the conclusion of the Mean Value Theorem.
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