Question

Suppose
*f(x)* is a function and that c is a critical number for
*f(x)* at which both *f**'* and
*f**''* are defined. Write a paragraph explaining
exactly what the Second Derivative Test allows us to conclude about
the point where x = c.

Answer #1

1. What is a critical number of a function f ? What is the
connection between critical numbers and relative extreme
values?
2. What does the sign of the derivative, f ', tell us about the
function? What does the sign of the 2nd derivative, f ", tell us
about the function?

Find the critical point of the function f(x,y)=x2+y2+xy+12x
c=________
Use the Second Derivative Test to determine whether the point
is
A. a local maximum
B. a local minimum
C. a saddle point
D. test fails

Problem 1.
(1 point)
Find the critical point of the function
f(x,y)=−(6x+y2+ln(|x+y|))f(x,y)=−(6x+y2+ln(|x+y|)).
c=?
Use the Second Derivative Test to determine whether it is
A. a local minimum
B. a local maximum
C. test fails
D. a saddle point

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)

Let f(x) = 4x^5 − 5x^4 + 9.
(a) Find each critical number for f.
(b) Apply the First Derivative Test to each critical number
found in part (a), being careful to explain what conclusions (if
any) you can draw.
(c) Apply the Second Derivative Test to each critical number
found in part (a), being careful to explain what conclusions (if
any) you can draw.

f(x)=x^3-4x^2+5x-2
Find all critical numbers of the function, then use the second
derivative test on each critical number to determine if it is a
local maximum or minimum. Show your work.

a) The function f(x)=ax^2+8x+b, where a and b are
constants, has a local maximum at the point (2,15). Find the values
of a and b.
b) if b is a positive constand and x> 0, find the
critical points of the function g(x)= x-b ln x, and determine if
this critical point is a local maximum using the second derivative
test.

Consider the following function.
f (x, y) = (x −
6) ln(x5y)
(a)
Find the critical point of f.
If the critical point is (a, b) then enter
'a,b' (without the quotes) into the answer
box.
(b)
Using your critical point in (a), find the value of
D(a, b) from the Second Partials test
that is used to classify the critical point.
(c)
Use the Second Partials test to classify the critical point
from (a).
one of: relative max, relative...

Consider the function f(x) =
x^2/x-1 with f ' (x) =
x^2-2x/ (x - 1)^2 and f ''
(x) = 2 / (x - 1)^3 are given. Use these to
answer the following questions.
(a) [5 marks] Find all critical points and determine the
intervals where f(x) is increasing and where it
is decreasing, use the First Derivative Test to fifind local
extreme value if any exists.
(b) Determine the intervals where f(x) is
concave upward and where it is...

f(x)=5x^(2/3)-2x^(5/3)
a. Give the domain of f
b. Find the critical numbers of f
c. Create a number line to determine the intervals on which f is
increasing and decreasing.
d. Use the First Derivative Test to determine whether each
critical point corresponds to a relative maximum, minimum, or
neither.

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