Question

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.

Answer #1

For each of the following questions, consider a
function, f(x) that is continuous on [a,b].
How would you find the critical values of f(x)?
Explain.
Where would f(x) be increasing/decreasing?
Explain.
At what possible x values would f(x) have extrema?
Explain.
Is it possible that f(x) is continuous and has no
extrema on the interval [a,b]? Use the Extreme Value Theorem to
explain your response.
If f’’(c) = 0, c in (a,b), and f’’(x) > 0 for all x
values...

Sketch the graph of a function that is
continuous on (−∞,∞) and satisfies the following sets of
conditions.
f″(x) > 0 on (−∞,−2); f″(−2) = 0; f′(−1) = f′(1) = 0; f″(2) =
0; f′(3) = 0; f″(x) > 0 on ( 4, ∞)

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].

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, ∞)...

If f is continuous on ( a , b ) and f ( x ) ≠ 0 for all x in ( a
, b ), then either f ( x ) > ______ for all x in ( a , b ) or f
( x ) < _________ for all x in ( a , b ).
A function f is said to be continuous on the _______ at x = c if
lim x → c +...

7. Sketch complete graphs of the following; indicate where the
function is increasing/decreasing, has local extrema, where the
graph is concave up or concave down, asymptotes and coordinate
intercepts
(a) ?(?)=(1/5)?5−? (b) ?(?)=(3?+2)/?

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

Sketch the graph of a function f(x) that satisfies all of the
conditions listed below. Be sure to clearly label the axes.
f(x) is continuous and differentiable on its entire domain,
which is (−5,∞)
limx→-5^+ f(x)=∞
limx→∞f(x)=0limx→∞f(x)=0
f(−2)=−4,f′(−2)=0f(−2)=−4,f′(−2)=0
f′′(x)>0f″(x)>0 for −5<x<1−5<x<1
f′′(x)<0f″(x)<0 for x>1x>1

For the function
f(x) =x(x−4)^3
•
Find all
x-intercepts and find the
y-intercept
•
Find all critical numbers,
•
Determine where the function is increasing and where it is
decreasing,
•
Find and classify the relative extrema,
•
Determine where the function is concave up and where it is
concave down,
•
Find any inflection points, and Use this information to sketch
the graph of the function.
•
Use this information to sketch the graph of the function.

For f(x) xe-x
( a) Find the local extrema by hand using the first derivative
and a sign chart. b) Find the open intervals where the function is
increasing and where it is decreasing. c) Find the intervals of
concavity and inflection points by hand. d) Sketch a reasonable
graph showing all this behavior . Indicate the coordinates of the
local extrema and inflections.

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