Question

In each part below, sketch a graph of a function whose domain is [0, 4] that has the desired property. No justification is needed in any part.

(a) f(x) has an absolute maximum and absolute minimum on [0, 4].

(b) g(x) has neither an absolute maximum nor an absolute minimum on [0, 4].

(c) h(x) has exactly two local minima and one local maximum on [0, 4].

(d) k(x) has one inflection point and no local extrema on [0, 4].

Answer #1

a)

On the interval [0,4], the absolute minimum value of f(x) is -5 at x = 0.

On the interval [0,4], the absolute maximum value of f(x) is 4 at x = 3.

Below is the graph of the function f(x).

b)

On the interval [0,4], g(x) has no absolute maximum and absolute minimum value as seen in the below graph.

c.

On the interval [0,4], h(x) has two local minima at x = 1 and x = 3 respectively.

On the interval [0,4], h(x) has one local maximum at x = 2.

Below is the graph of the function h(x).

d.

On the interval [0,4], k(x) has one inflection point at x = 2 and no local extrema.

Below is the graph of the function k(x).

sketch graph of a function that continuous on [1,5]
an absolute minimum value at x=4
an absolute maximum value at x=5
a local minimum at x=2
a local maximum at x=3

Sketch the graph of f by hand and use your sketch to
find the absolute and local maximum and minimum values of
f. (If an answer does not exist, enter DNE.)
f(x) =
25 − x2
if −5 ≤ x < 0
4x − 2
if 0 ≤ x ≤ 5
absolute maximum
absolute minimum
local maximum
local minimum

For the following exercises, draw a graph that satisfies the
given specifications for the domain x=[−3,3]. The function does not
have to be continuous or differentiable.
216.
f(x)>0,f′(x)>0 over x>1,−3<x<0,f′(x)=0 over
0<x<1
217.
f′(x)>0 over x>2,−3<x<−1,f′(x)<0 over
−1<x<2,f″(x)<0 for all x
218.
f″(x)<0 over
−1<x<1,f″(x)>0,−3<x<−1,1<x<3, local maximum at
x=0, local minima at x=±2
219.
There is a local maximum at x=2, local minimum at x=1, and the
graph is neither concave up nor concave down.

Sketch the graph of f by hand and use your sketch to
find the absolute and local maximum and minimum values of
f. (Enter your answers as a comma-separated list. If an
answer does not exist, enter DNE.)
f(x) = 1/4(5x-2) . x ≤ 3
absolute maximum value
absolute minimum value
local maximum value(s)
local minimum value(s)

Sketch the graph of a function f that is continuous on (−∞,∞)
and has all of the following properties:
(a) f0(1) is undeﬁned
(b) f0(x) > 0 on (−∞,−1)
(c) f is decreasing on (−1,∞).
Sketch a function f on some interval where f has one inﬂection
point, but no local extrema.

Curve Sketching Practice
Use the information to the side to sketch the graph of
f.
Label any asymptotes, local extrema, and inflection
points.
f is a polynomial function
x
—1
—6
3
—2
6
5
f is a polynomial function
x
1
—4
4
0
7
4

Analyze and sketch the graph of the function f(x) = (x −
4)2/3
(a) Determine the intervals on which f(x) is increasing /
decreasing
(b) Determine if any critical values correspond to a relative
maxima / minima
(c) Find possible inflection points
(d) Determine intervals where f(x) is concave up / down

1) The function f(x)=2x3−33x2+108x+3f(x)=2x3-33x2+108x+3 has one
local minimum and one local maximum. Use a graph of the function to
estimate these local extrema.
This function has a local minimum at x
= with output value =
and a local maximum at x = with output
value =
2) The function f(x)=2x3−24x2+42x+7 has one local minimum and
one local maximum. Use a graph of the function to estimate these
local extrema.
This function has a local minimum at x
= with output value =...

Use analytical methods to find all local extrema of the function
f(x)=3x^1/x for x>0.
The function f has an absolute maximum of ? at x=?
The function f has an absolute minimum of ? at x=?

Sketch a graph of a function having the following properties.
Make sure to label local extremes and inflection points.
1) f is increasing on (−∞, −2) and (3, 5) and decreasing on (−2,
0),(0, 3) and (5,∞).
2) f has a vertical asymptote at x = 0.
3) f approaches a value of 1 as x → ∞
4) f does not have a limit as x → −∞
5) f is concave up on (0, 4) and (8, ∞)...

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