Question

Under what circumstances can you be certain that an absolute maximum and absolute minimum will exist for a function? (Hint: this has to do with limiting the function to a certain kind of set.) Give an example of a subset of R2 that fulfills the conditions, and an example of one that doesn’t. Describe how you find the maximum and minimum values of a function on such a set.

Answer #1

There is a well-known theorem for absolute maxima and minim, states that:

If f **is** continuous on a closed
**interval** [a, b], then f **has** both
an **absolute maximum value and an absolute minimum
value** on the **interval**. This theorem says
that a continuous **function** that
**is** defined on a closed **interval must
have** both an **absolute maximum value and an
absolute minimum value**.

Find the absolute maximum and absolute minimum values, if any,
of the function. (If an answer does not exist, enter DNE.)
h(x) = x3 + 3x2 + 3 on [−3, 2]
absolute maximum ______
absolute minimum _______

Find the absolute maximum and absolute minimum values (if they
exist) of f(x) = ln x/x on the interval (0, ∞).

Exercise 9.
In the questions below you can describe the relations/functions
either by drawing a diagram, by a formula, or by listing the
ordered pairs. Explain your solutions.
(i) Give an example of two sets A and B and a relation R from A to
B which is not a function.
(ii) [hard] Find a set A, |A| = 4 and deﬁne a bijective function
between A and P(A)? If such a set doesn’t exist give a reason.
Exercise 11....

find the absolute maximum value and absolute minimum values of
the function f(x,y)4xy^2-x^2y^2-xy^3 on the set D, where D is the
closed trianglar region in the xy-plane with certices
(0,0)(0,6)(6,)0

Find the absolute maximum and minimum values of the following
function on the given interval. If there are multiple points in a
single category list the points in increasing order in x value and
enter N in any blank that you don't need to use. f(x)=3(x2−1)3,
[−1,2]
Absolute maxima
x = y =
x = y =
x= y =
Absolute minima
x = y =
x = y =
x = y =

Find the absolute maximum and minimum values of the following
function over the given interval. If there are multiple points in a
single category list the points in increasing order in x value and
enter N in any blank that you don't need to use.
?(?)=(10cos?)/(18+9sin?), 0≤?≤2?

Find the absolute maximum and minimum values of the following
function over the given interval. If there are multiple points in a
single category list the points in increasing order in x value and
enter N in any blank that you don't need to use.
f(x)=(4cosx)/(14+7sinx), 0≤x≤2π
Absolute maxima
x = y =
x = y =
x = y =
Absolute minima
x = y =
x = y =
x = y =

1. Find the absolute maximum and minimum values of each function
on the given interval SHOW WORK
f(x)=√(16-x^2 ),0≤x≤3
2. Maximum Height of Vertically Moving
Body: The height of a body moving vertically is given
by s=-4.9t2+5t+10 with s in
meters and t in seconds. Find the body’s maximum height along with
the time. SHOW WORK. Give answers with two (2) decimal places. SHOW
WORK
3. Checking the Mean Value Theorem:
Find the value or values of c that satisfy...

Show all of your work neatly
whenever possible. Justify your answers.
10. Find the absolute maximum and absolute minimum values of
f(x)= x+2 cosx on the interval [0, 2π] Give
exact answers. You can use decimals to 3 places to
approximate the values of f(x) . (4 points)
Absolute Max: ______________
Absolute Min: ______________

Find the local maximum and minimum values and saddle point(s) of
the function. If you have three-dimensional graphing software,
graph the function with a domain and viewpoint that reveal all the
important aspects of the function. (Enter your answers as a
comma-separated list. If an answer does not exist, enter DNE.)
f(x, y) =
4y
cos(x), 0 ≤
x ≤ 2π
Find:
local maximum value(s) =
local minimum value(s) =
saddle point(s) (x,y,f) =

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