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Review: 3.1-3.3 on MVT and Derivative Graphing
***No Calculator on quiz (you will not need it!)
Steps for the Mean Value Theorem:
Steps for finding when f(x) is increasing, decreasing, and the critical points (max/min):
Steps for finding when f(x) is concave up, concave down, and the inflection points:
1. Determine if the following functions satisfy the hypothesis for the Mean Value Theorem on the
interval given. If they do, determine the exact values of c that satisfy the MVT.
a. f ( x)  4 x 2  x  6 on [1, 3]
b. p( x)  ( x  1)3 on [0, 1]
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d. h( x)  2 x  1 on [1, 5]
c. g ( x)  x3 on [1, 3]
2. Find the critical points for each of the following functions, then determine where they are increasing
and decreasing, and classify all extreme values (max’s and min’s).
a. f ( x)  3x 2  5 x  1
b. h( x)  x 4  2 x 2  3
c. f ( x)  ( x  3)2 ( x  1)
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3. Determine the intervals where the following functions are concave up or down, and find any
inflection points.
a. g ( x)  x3  5 x
b. h( x)  (3x  2)3
*c. f '( x)  x 2  3x  4
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4.
Sketch the graph of a function that satesfies the following characteristics.
a.
5.
b.
𝑓(𝑥) < 0 𝑜𝑛 (3, ∞), 𝑓(𝑥) <
0 𝑜𝑛 (−∞, 3), 𝑓 ′ (𝑥) >
0 𝑜𝑛 (−∞, 2) ∪ (2, ∞),
𝑓 ′′ (𝑥) < 0 𝑜𝑛 (−∞, 2), and
𝑓 ′′ (𝑥) > 0 𝑜𝑛 (2, ∞) .
For each of the following functions, find the zeros, y-intercept, coordinates of local extrema,
intervals where the function increases and/or decreases, coordinates of inflection points,
intervals where the function is concave up and/or down, and then sketch the graph.
a.
𝑓(𝑥) = 𝑥 4 + 4𝑥
b.
𝑓(𝑥) = (𝑥 − 2)(𝑥 + 1)2
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