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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 - x2 + 3. Use the Second Derivative Test to determine which critical numbers, if any, give relative extrema. x = 0, relative maximum; relative minimum Review Problems 20

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

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

– 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

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