1. Review Problems Sections 3-1 to 3-4
Calculus 3-R-a

2. Find all critical numbers for the functions:
– 3
Review Problems 1

3. Review Problems 2 f(x) = 3x4 - 4x3 x = 0, 1
Find all critical numbers for the functions:
f(x) = 3x4 - 4x3
x = 0, 1
Review Problems 2

4. Review Problems 3 f(x) = (x + 2)4(x - 1)3 1 f(x) = x3(x + 3)2 and 0
Find all critical numbers for the functions:
f(x) = (x + 2)4(x - 1)3
-2, 1
f(x) = x3(x + 3)2
-3,
and 0
Review Problems 3

5. Review Problems 4 Find all extrema in the interval [0, 2π] for
y = x + sin x.
Minimum: (0, 0)
Maximum: (2π, 2 π)
Find the absolute maximum and absolute minimum of f on the interval (0, 3].
Maximum: None
Minimum: (2, 3)
Review Problems 4

6. Find the absolute maximum and absolute minimum of f on the interval (1, 4].
Maximum: None
Minimum: (3, -3)
Find the absolute maximum and absolute minimum of f on the interval (-4, -1].
Maximum: None
Minimum: (-2, -1)
Review Problems 5

7. Find the value of the derivative (if it exists) at the indicated extremum.
Determine from the graph whether f possesses extrema on the interval (a, b).
Maximum at x = c,
no minimum
Review Problems 6

8. Review Problems 7 Consider f is not continuous on [2, 4]
Sketch the graph of f(x).
Calculate f(2) and f(4).
c. State why Rolle’s Theorem does not apply to f on the interval [2, 4].
f(2) = f(4) = 1
f is not continuous on [2, 4]
Review Problems 7

9. Review Problems 8 Decide whether Rolle’s Theorem can be applied to
on the interval [-2, 0]. If Rolle’s Theorem can be applied, find all value(s), c, in the interval such that
If Rolle’s Theorem cannot be applied, state why.
Rolle’s Theorem does not apply because f(-2) ≠ f(0).
Review Problems 8

10. Given
find all c in the interval (2, 8) such that 4
Review Problems 9

11. Determine whether the Mean Value Theorem applies to f(x) = 3x - x2 on the interval [2, 3]. If the Mean Value Theorem can be applied, find all value(s) of c in the interval such that
If the Mean Value Theorem does not apply, state why.
The Mean Value Theorem applies;
Review Problems 10

12. Review Problems 11 Determine whether the Mean Value Theorem applies to
on the interval [-1, 3]. If the Mean Value Theorem can be applied, find all value(s) of c in the interval such that
If the Mean Value Theorem does not apply, state why.
The Mean Value Theorem does not apply because f is not continuous at
Review Problems 11

13. Review Problems 12 (-, 0) Find all open intervals on which
is decreasing.
(-, 0)
Find the open intervals on which
is increasing or decreasing.
Increasing (- , 0);
decreasing (0, )
Review Problems 12

14. Find the values of x that give relative extrema for the function f(x) = 3x5 - 5x3
Relative maximum: x = -1;
relative minimum: x = 1
Find the values of x that give relative extrema for the function f(x) = (x + 1)2(x - 2)
Relative maximum: x = -1;
relative minimum: x = 1
Review Problems 13

15. Use the first derivative test to find the x-values that give relative extrema for f(x) = -x4 + 2x3
Relative maximum at
Find the relative minimum and relative maximum for f(x) = 2x3 + 3x2 - 12x
Relative maximum: (-2, 20); relative minimum: (1, -7)
Review Problems 14

16. Review Problems 15 Relative maximum
Let
Show that f has no critical numbers
≠0 for all x ≠ 1.
is undefined at x = 1, a vertical asymptote
A differentiable function f has only one critical number: x = -3. Identify the relative extrema of f at (-3, f(-3)) if
and
Relative maximum
Review Problems 15

17. Find all intervals on which the graph of the function is concave upward:
(-, 0) and (0, )
Find all intervals on which the graph of the function is concave upward:
(- , -3)
Review Problems 16

18. Find all intervals for which the graph of the function y = 8x3 - 2x4 is concave downward
(- , 0) and (2, )
Find the intervals on which the graph of the function f(x) = x4 - 4x3 + 2 is concave upward or downward. Then find all points of inflection for the function
Concave upward: (- , 0), (2, )
Concave downward: (0, 2)
Points of inflection: (0, 2) and (2, -14)
Review Problems 17

19. Find all points of inflection of the graph of the function f(x) = x3 - 12x.
(0, 0)
Find all points of inflection of the graph of the function f(x) = 2x(x - 4)3
(4, 0), (2, -32)
Review Problems 18

20. Find all points of inflection of the graph of the function f(x) = x4 - 6x3
(0, 0) and (3, -81)
Let
and let f(x) have critical numbers -1, 0, and 1. Use the Second Derivative Test to determine which critical numbers, if any, give a relative maximum. -1
Review Problems 19

21. Let
and let f(x) have critical numbers -2, 0, and 2. Use the Second Derivative Test to determine which critical numbers, if any, give a relative maximum.
Let f(x) = x3 - x Use the Second Derivative Test to determine which critical numbers, if any, give relative extrema.
x = 0, relative maximum;
relative minimum
Review Problems 20

22. Find all relative extrema of the function f(x) = x4 + 4x3
Find all relative extrema of the function f(x) = x4 + 4x3. Use the Second Derivative Test where applicable
Relative minimum: (-3, -27)
Give the sign of the second derivative of f at the indicated point
Positive
Review Problems 21

23. The graph of a polynomial functions, f, are given
The graph of a polynomial functions, f, are given. On the same coordinate axes sketch
and
Review Problems 22

24. – 3
The Mean Value Theorem does not apply because f is not continuous at
Increasing (- , 0);
decreasing (0, )
(-, 0)
x = 0, 1
-2, 1
-3,
and 0
Relative maximum: x = -1;
relative minimum: x = 1
Relative maximum: x = -1;
relative minimum: x = 1
Minimum: (0, 0)
Maximum: (2π, 2 π)
Maximum: None
Minimum: (2, 3)
Relative maximum at
Relative maximum: (-2, 20); relative minimum: (1, -7)
Maximum: None
Minimum: (3, -3)
Maximum: None
Minimum: (-2, -1)
Maximum at x = c,
no minimum
≠0 for all x ≠ 1.
is undefined at x = 1, a vertical asymptote
f(2) = f(4) = 1
f is not continuous on [2, 4]
Relative maximum
(-, 0) and (0, )
(- , -3)
Rolle’s Theorem does not apply because f(-2) ≠ f(0). 4
(- , 0) and (2, )
Concave upward: (- , 0), (2, )
Concave downward: (0, 2)
Points of inflection: (0, 2) and (2, -14)
The Mean Value Theorem applies;
Answers

25. Answers Graph (0, 0) and (3, -81) -1 (0, 0) (4, 0), (2, -32)
x = 0, relative maximum;
relative minimum
Relative minimum: (-3, -27)
Positive
Answers