Name:______________________________
Date: __________
Unit 3 Review
Applications of Derivatives
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Period:________
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9. The derivative of f is x3  x  5 x 1 . At how which values of x will the graph of f have a relative
maximum? A relative minimum?
10. If f ''  x    x  5  x  1 x  2  , then the graph of f has inflection points when x  ?
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2x2
on  5,5 .
x2  4
a. On what interval(s) is f increasing? Justify your conclusion.
11. Let f ( x) 
b. On what interval(s) is f concave up? Justify your conclusion.
c. At what value(s) of x does f have an inflection point? Justify your conclusion.
12. Let f be the function given by f  x    x 3 . Which of the following statements about f are true?
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I. f is continuous at x  0 .
II. f is differentiable at x  0 .
III. f has an absolute maximum at x  0 .
y  f ( x)
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y
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The graph of the derivative of the function, f ( x) , is shown at the right.
a. Locate the x-value(s) where there is a relative maximum. Justify.
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b. Locate the x-value(s) where there is a relative minimum. Justify.
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14. The figure to the right shows the graph of f  ,
the derivative of the function f on the closed interval
2  x  8 . The graph of f  has horizontal tangents
at x  1 and x  5 . The function is twice
differentiable with f (3)  2.
a. Find the x-coordinate of the point(s) of inflection
of the graph of f. Give a reason for your answer.
b. For what values of x does f attain its absolute
maximum value on the closed interval 2  x  8 ?
Show the analysis that leads to your answer.
c. Using the known points given on the graph of f  , for what value(s) of x does the graph of
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y   x 2  f ( x) have a horizontal tangent? Give a reason for your answer.
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d. Let g be the function defined as g ( x)  x 2 f ( x) . Find an equation for the line tangent to the graph of g
at x  3.
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15. Find the values of c that satisfy Rolle’s Theorem for 𝑓(𝑥) = 𝑥 2 − 8𝑥 + 12 on the interval [2, 6].
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