4.2-What derivatives tell us

Question 1

A function F is increasing if

Answer 1

If f’(x) > 0 at all interior points of I, then f is increasing on I

Question 2

A function is decreasing if

Answer 2

If f’(x)

Question 3

Find the intervals on which the function f(x) = 2x^3 + 3x^2 + 1

Answer 3

critical points are 0 and -1
f is increasing on (-infin, -1
f is decreasing on (-1,0)
f is increasing on (0, infin)

Question 4

First derivative Test

Answer 4

If f’ changes signs from positive to negative through c, then f has a local MAXIMUM at C

If f’ changes sign from negative to positive as x increases through c, then f has a local MINIMUM at c

If f’ does not change sign at c then f has no local extreme value at c

Question 5

f(x) = 3x^4 - 4x^3 - 6x^2 +12x + 1

a. find intervals on which f is increasing and decreasing
b. identify the local extrema of f

Answer 5

g

Question 6

Concavity and inflection points

Answer 6

If f is continuous at c and f changes concavity at c (from up to down, or vice versa), then f has an inflection point at c

Question 7

Test for concavity

Answer 7

if f” > 0 on i, then f is concave up on I

if f”

Question 8

Sketch a function satisfying each set of conditions
f’(t) > 0 and f” (t) > 0

b. g’(t) > 0 and g”(t)

Answer 8

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Question 9

2nd derivative test for local extreme

Answer 9

Suppose f’(c) = 0
thennnnn

If f”(c) > 0, then f has local minimum at c

If f”(c)

Question 10

Recap of derivative properties

Answer 10

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