5.3-5.7

Question 1

critical value

Answer 1

where f’(x) is zero or undefined

Question 2

relative extrema only occur at…

Answer 2

critical values
if f(c) is a relative extrema, then c is a CV

Question 3

how to find extrema on a closed interval

Answer 3

1. find CVs on the interval
2. evaluate the CVs in the original equation
3. evaluate the endpoints of the interval in the original equation
4. the smallest number = abs min and vice versa

Question 4

is the x or y value the extrema?

Answer 4

the actual extrema is the y value, the x value is where the extrema occurs

Question 5

how to determine where a function in increasing and decreasing

Answer 5

1. if f’(x)>0 then f(x) is increasing
2. if f’(x)<0 then f(x) is decreasing
3. if f’(x)=0 then f(x) is constant (poss extrema)

Question 6

steps to determining intervals where functions are inc/dec

Answer 6

1. find all CVs of f(x)
2. make a sign chart with CVs and test points into f’(x)
3. where f’(x)>0, f(x) is inc and vice versa

Question 7

first derivative test

Answer 7

1. find f’(x)
2. set f’(x) = 0
3. make a sign chart with zeros
4. fill in sign chart using f’(x)
5. apply rules

Question 8

if f’(x) changes from negative to positive at c then f(c) is a

Answer 8

relative minimum

Question 9

if f’(x) changes from positive to negative at c then f(c) is a

Answer 9

relative maximum

Question 10

if f’(x) does not change sign at c then f(c) is

Answer 10

neither a max nor a min

Question 11

slopes vs. concavity

Answer 11

CU = increasing slopes
CD = decreasing slopes

Question 12

test for concavity

Answer 12

if f”(x)>0 the f(x) is CU and vice versa

Question 13

point of inflection

Answer 13

where a function changes concavity

Question 14

point of inflection theorem

Answer 14

if the point (c, f(c)) is a POI, then f”(c) will be a CV

Question 15

the second derivative test

Answer 15

1. find where f’(c)=0 (CVs)
2. if f”(c)>0 then f(c) is a minimum because f(x) is CU at x=c (and vice versa)