house

what is the order of the calculus

+ to - increasing to decreasing concave up to concave down

How to find absolute extrema on a closed interval:

1. Evaluate f(x ) at endpoints 2. Evaluate f(x ) at critical numbers. 3. The largest value is the absolute max and the smallest value is the absolute min.

How to find relative extrema - the Second Derivative Test:

1. Find all points x where f'(x ) =0 . 2. Find values of f "(x ) for these x values. 3. If f"(x ) > 0, then f (x ) is a relative min. If f "(x) ) < 0, then f(x ) is a relative max. If f"(x )= 0, then do first derivative test.

definition of an inflection point

1.)f"(c)=0 or if f"(c)= undefined and 2.) if f" changes from pos to neg or neg to pos at x=c or if f' changes from inc to dec or dec to incat x=c

If f"(c)=0 and f;(x) decreases through x 5 c, then x 5 c locates a local minimum value for the function.

11. False Counterexample: For , , decreases through x 5 0, and f has a local (and global) maximum value at x 5 0. Rewrite: If and decreases through x 5 c, then x 5 c locates a local MAXIMUM value for the function. Or, if and INCREASES through x 5 c, then x 5 c locates a local minimum value for the function.

. If 𝑓 ′ (𝑥) changes from negative to positive at 𝑥 = 𝑐, then 𝑥 = 𝑐 -----------------locates a critical point of 𝑓(𝑥).

ALWAYS

. If 𝑓′ (𝑥) > 0 for [𝑎, 𝑏], and 𝑓"(𝑥) changes from positive to negative at 𝑥 = 𝑐, then 𝑓(𝑥) ---------------changes from increasing at an increasing rate to increasing at a decreasing rate

ALWAYS

. If 𝑓′(𝑥)changes from decreasing to increasing at 𝑥 = 𝑐, then 𝑥 = 𝑐 ---------------locates a point of inflection for 𝑓(𝑥).

ALWAYS

To locate the absolute extrema of a continuous function on a closed interval, you need only compare the y-values of all critical points.

False Counterexample: #e function on the interval 2 # x # 6 has its absolute minimum value at x 5 2 and its absolute maximum value at x 5 6. Neither x 5 2 nor x 5 6 is a critical point of the function. Rewrite: To locate the absolute extrema of a continuous function on a closed interval, you must compare the y-values of all critical points AND ENDPOINTS.

the absolute minimum value of a continuous function on a closed interval can occur at only one point

False Counterexample: #e function takes on its minimum value of 21 at the points in the interval 0 , x , 6π. Rewrite: #ere is exactly one absolute minimum value of a continuous function on a closed interval, but this minimum value can occur at more than one point in the interval. See Problem 2.

If a function f has a local minimum value at x 5 c, then f'(c)=0

False Counterexample: e function has a local minimum at x 5 0, but is not de!ned. Rewrite: If a DIFFERENTIABLE function has a local minimum value at x 5 c, then .

. If 𝑥 = 𝑐 is the only critical point in the interval [𝑎, 𝑏] and 𝑓 ′ (𝑥) changes from postive to negative at 𝑥 = 𝑐, then 𝑥 = 𝑐 -------------locates an absolute minimum of 𝑓(𝑥).

NEVER

If 𝑓 ′ (𝑥) < 0 for [𝑎, 𝑏], then 𝑥 = 𝑏 --------------locates an absolute maximum of 𝑓(𝑥).

NEVER

if f'(x) changes from increasing to decreasing f'(x) has

R max

. If 𝑓 ′ (𝑥) = 0 at 𝑥 = 𝑐, then 𝑥 = 𝑐 ----------------locates a either a relative maximum or relative minimum for 𝑓(𝑥)

SOMETIMES

If 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) > 0, then 𝑥 = 𝑐 ----------------locates an absolute minimum of 𝑓(𝑥).

SOMETIMES

If 𝑓"(𝑐) = 0, then 𝑥 = 𝑐 -------------locates a point of inflection of 𝑓(𝑥)

SOMETIMES

If 𝑓"(𝑐) > 0, then 𝑥 = 𝑐 --------------locates a local minimum of 𝑓(𝑥)

SOMETIMES

If x = 2 is the only critical point of a function f and f"(2)=3, then x = 2 locates a ____________________ value of the function.

abs minimum

If x = 2 is the only critical point of a function f and f"(2)<0, then x = 2 locates __________________ value of the function.

absolute maximum

If a continuous function f increases throughout a closed interval, then the le" endpoint of the graph of f on the interval is _________________________ point of the function

absolute minimum

f'(x) is decreasing f(x)=

concave down

if f"(x)<0 for all x in an open interval the the graph is

concave down

f(x) is concave down f'(x)=

decreasing

IF F'(X) > 0 then f is concave upward at c

false

f(x) is concave up f'(x)=

increasing

the graph of f is concave upward on an interval if f' is---------- on the interval

increasing

f''(x) changes from + to - f(x)=

inflection point

f''(x) changes from - to + f(x)=

inflection point

f'(x) has a max f(x)=

inflection point

1.)f"(c)=0 or if f"(c)= undefined and 2.) if f" changes from pos to neg or neg to pos at x=c

point of inflection

2. A continuous function on a closed interval can have only one maximum value.

true

Absolute extrema of a continuous function on a closed interval can occur only at endpoints or critical points.

true

If x 5 2 is the only critical point of a function f and f"(2)>0, then f(2) is the minimum value of the function

true

when is there an inflection point

when it goes from concave up to concave down or vise versa when f''(x) changes - to + or vise versa when f'(x) has a minimum or maximum

If 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) < 0, then 𝑥 = 𝑐 ---------------locates a local maximum of 𝑓(𝑥).

ALWAYS

If 𝑓 ′ (𝑥) > 0 for [𝑎, 𝑏], then 𝑥 = 𝑎 ---------------locates an absolute minimum of 𝑓(𝑥).

ALWAYS

If 𝑓"(𝑥)changes signs at 𝑥 = 𝑐, then 𝑥 = 𝑐 ------------------locates a point of inflection for 𝑓(𝑥)

ALWAYS

1. If 𝑓 ′ (𝑥) changes from negative to positive at 𝑥 = 𝑐, then 𝑥 = 𝑐 -----------------------locates a relative minimum of 𝑓(𝑥)

ALWAYS

f(x) has a max f'(x)=

0

f(x) has a min f'(x)=

0

How to find points of inflection:

1. Determine concavity described above. 2. Where f"(x ) changes signs will be an inflection point.

How to find relative extrema - the First Derivative Test:

1. Find critical numbers, put them on a number line, and sign test in f'(x ). 2. If f'(x) changes from negative to positive, you have a relative minimum. If f'(x) changes from positive to negative, you have a relative maximum.

How to tell if function is increasing or decreasing:

1. Find critical numbers. 2. Put critical numbers on a number line and test in f'(x ) . Positive interval indicates increasing, negative interval indicates decreasing.

How to determine concavity:

1. Set f"(x ) 0 =0 and solve for x. 2. Find out where f"(x) is undefined. 3. Use these xvalues to do sign testing in f"(x) . If f"(x ) > 0, the curve is concave upward. If f"(x) < 0, the curve is concave downward.

How to find critical numbers:

1. Set f'(x)=0 and solve for x. 2. Find values where f'(x ) is undefined.

. If 𝑥 = 𝑐 is the only critical point in the interval [𝑎, 𝑏] and 𝑓 ′ (𝑐) = 0 and 𝑓"(𝑐) < 0, then 𝑥 = 𝑐 ------------------locates an absolute maximum of 𝑓(𝑥).

ALWAYS

. If 𝑥 = 𝑐 is the only critical point in the interval [𝑎, 𝑏] and 𝑓 ′ (𝑥) changes from positive to negative at 𝑥 = 𝑐, then 𝑥 = 𝑐 ----------locates an absolute maximum of 𝑓(𝑥).

ALWAYS

If f"(x) is always positive, then the function f must have a relative minimum value

False Counterexample: For , is always positive, but the function has no relative extrema. Rewrite: If and is always positive, then the function must have a relative minimum at x 5 c.

If f"(2)<0, then x = c is a point of in#ection for the function f and cannot be the x-coordinate of a maximum or minimum point on the graph of f.

False Counterexample: For the function , , but x 5 0 is not a point of in!ection. Note that x 5 0 does correspond to a relative and absolute minimum value of f. Rewrite: If for a function f, then x 5 c may or may not be an in!ection point for f and x 5 c may or may not correspond to a relative minimum or maximum value of f.

A critical point (or critical number) of a function f of a variable x is the x-coordinate of a relative maximum or minimum value of the function.

False Counterexample: Note that x 5 0 is a critical point for the function , but that x 5 0 corresponds to neither a relative maximum nor a relative minimum value of f. Rewrite: A critical point is a POSSIBLE location for a relative maximum or minimum value of a function.

if f'(x) changes from decreasing to increasing f'(x) has

R min

f'(x) changes from + to - f(x)=

maximum

f'(x) changes from - to + f(x)=

minimum

f'(x) is increasing f(x)=

concave up

if f"(x)>0 for all x in an open interval the the graph is

concave up

the graph of f is concave downon an interval if f' is---------- on the interval

decreasing

Rolle's Theorem can be applied to f(x) = 1/x^2 on the interval [-1,1]

false

at each critical number of a function there is a relative max or min

false

if a function is continous on a closed interval then it must have a minimum or a maximum but not both on the iterval

false

if a function is continous on a closed interval then it must have a rel max on the interval

false

if the graph of a function has three x intercepts then it must have at least two relative extrema

false

if x=c is a critical number of the function f then it is also a critical number of the function g(x) where g(x) =f(x-k)

false

the relative maxima of the function f are f(1)=4 and f(3)=10 therefore f has at least one relative minimum for some x in the interval (1,3)

false

if f'(c)>0 then f is concave upward at c

false concavity is determined by f". for example let f(x) = x and c=2 f'(c) = d'(2) >0 but f is not concave upward at 2

if f"(2)=0 the graph of f must have a point of inflection at x=2

false could not change sign

the graph f(x) = 1/x is concave downward for x<0 and concave upward for x> 0 and thus has a pointof inflection

false it has a discontinuity at x=0

f'(x) has a min f(x)=

inflection point

if f(x) changes from concave up to concave down f(x) has

inflection point

If x = 2 is a critical point of the function f, and f'(x) decreases through x = 2, then x 5 2 locates _________________________ value of the function.

local maximum

If f'(2) =0 and f"(2)>0 , then x = 2 locates _____________________ value of the function f.

local minimum

If f'(2)=0 and f'(x) changes from negative to positive at x 5 2, then x 5 2 locates ____________________ value of the function f

local minimum

f (x) is decreasing f'(x)=

negative

f'(x) is decreasing f''(x)=

negative

f(x) is concave down f''(x)=

negative

f (x) is increasing f'(x)=

positive

f'(x) is increasing f''(x)=

positive

f(x) is concave up f''(x)=

positive

if f"(c)=0 and f"(c)<0 then f(c) is a --------------- of f

rel max

if f'(c)=0 and f"(c)>0 then f(c) is a --------------- of f

rel min

If a function f is de!ned on a closed interval and f;(x)>0 for all x in the interval, then the absolute maximum value of the function will occur at the right endpoint of the interval.

true

The graph of every cubic polynomial has precisely one point of inflection

true

if f'(2)=0 and f"(2)<0 If a function f has a local max value at x 5 c, then .

true

if the graph of a polynomial function has two x-intercepts then it must have at leastone point at which its tangent line is horizontal

true

if x=c is a critical number of a function f then it is also a critical number of the function g(x) where g(x)=f(x) + k

true

the maximum of a continous function on a closed interval can occur at two different values in the interval

true