Question

Given the function g(x)=4x3−24x2−60xg(x)=4x3-24x2-60x, find the
first derivative, g'(x)g′(x).

g'(x)=g′(x)=

Notice that g'(x)=0g′(x)=0 when x=−1x=-1, that is,
g'(−1)=0g′(-1)=0.

Now, we want to know whether there is a local minimum or local
maximum at x=−1x=-1, so we will use the second derivative
test.

Find the second derivative, g''(x)g′′(x).

g''(x)=g′′(x)=

Evaluate g''(−1)g′′(-1).

g''(−1)=g′′(-1)=

Based on the sign of this number, does this mean the graph of
g(x)g(x) is concave up or concave down at x=−1x=-1?

[Answer either **up** or **down** --
watch your spelling!!]

At x=−1x=-1 the graph of g(x)g(x) is concave

Based on the concavity of g(x)g(x) at x=−1x=-1, does this mean that
there is a local minimum or local maximumat x=−1x=-1?

[Answer either **minimum** or **maximum**
-- watch your spelling!!]

At x=−1x=-1 there is a local

Answer #1

Given the function g(x)=4x^3−24x^2+36x, find the first
derivative, g'(x)
g′(x)= ???
Notice that g'(x)=0 when x=1, that is, g'(1)=0
Now, we want to know whether there is a local minimum or local
maximum at x=1, so we will use the second derivative test.
Find the second derivative, g''(x)
g''(x)=????
Evaluate g"(1)
g''(1)=???
Based on the sign of this number, does this mean the graph of g(x)
is concave up or concave down at x=1?
[Answer either up or down --...

Consider the following. f(x) = 4x3 − 6x2 − 24x + 4
(a) Find the intervals on which f is increasing or decreasing.
(Enter your answers using interval notation.) increasing
decreasing
(b) Find the local maximum and minimum values of f. (If an
answer does not exist, enter DNE.) local minimum value local
maximum value
(c) Find the intervals of concavity and the inflection points.
(Enter your answers using interval notation.)
concave up concave down inflection point (x, y) =

1) Use the First Derivative Test to find the local maximum and
minimum values of the function. (Enter your answers as a
comma-separated list. If an answer does not exist, enter DNE.):
g(u) = 0.3u3 + 1.8u2 + 146
a)
local minimum values:
b)
local maximum values:
2) Consider the following:
f(x) = x4 − 32x2 + 6
(a) Find the intervals on which f is increasing or
decreasing. (Enter your answers using interval notation.)
increasing:
decreasing:...

Given the function g(x) = x3-3x + 1, use the first and second
derivative tests to
(a) Find the intervals where g(x) is increasing and
decreasing.
(b) Find the points where the function reaches all realtive
maxima and minima.
(c) Determine the intervals for which g(x) is concave up and
concave down.
(d) Determine all points of inflection for g(x).
(e) Graph g(x). Label your axes, extrema, and point(s) of
inflection.

1. Use the first derivative test and the second derivative test
to determine where each function is increasing, decreasing,
concave up, and concave down.
y=20x e^-x , x>0
2. Use the first derivative and the second derivative test to
determine where each function is increasing, decreasing, concave
up, and concave down.
y=4sin(πx^2), 10≤x≤1

1.) Suppose g(x) = x2− 3x.
On the interval [0, 4], use calculus to identify x-coordinate of
each local / global minimum / maximum value of g(x).
2.) For the function f(x) = x 4 − x 3 + 7...
a.) Show that the critical points are at x = 0 and x = 3/4 (Plug
these into the derivative, what you get should tell you that they
are critical points).
b.) Identify all intervals where f(x) is increasing
c.)...

f(x)= (x^2+2x-1)/x^2)
Find the
a.) x-intercept
b.) vertical and horizontal asymptote
c.) first and second derivative
d.) Is it increasing or decreasing? Identify any local
extrema
e.) Is it concave up and down? Identify any points of
reflection.

Given the function
h(x)=e^-x^2
Find first derivative f ‘ and second derivative
f''
Find the critical Numbers and determine the intervals
where h(x) is increasing and decreasing.
Find the point of inflection (if it exists) and determine
the intervals where h(x) concaves up and concaves
down.
Find the local Max/Min (including the
y-coordinate)

1. At x = 1, the function g( x ) = 5x ln(x) −
3x
is . . .
Group of answer choices
has a critical point and is concave up
decreasing and concave up
decreasing and concave down
increasing and concave up
increasing and concave down
2. The maximum value of the function f ( x ) = 5xe^−2x
over the domain [ 0 , 2 ] is y = …
Group of answer choices
10/e
0
5/2e
e^2/5...

consider the function
f(x)=3x-5/sqrt x^2+1. given f'(x)=5x+3/(x^2+1)^3/2 and
f''(x)=-10x^2-9x+5/(x^2+1)^5/2
a) find the local maximum and minimum values. Justify your
answer using the first or second derivative test . round your
answers to the nearest tenth as needed.
b)find the intervals of concavity and any inflection points of
f. Round to the nearest tenth as needed.
c)graph f(x) and label each important part (domain, x- and y-
intercepts, VA/HA, CN, Increasing/decreasing, local min/max values,
intervals of concavity/ inflection points of f?

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