Question

(a) Find the maximum and minimum values of f(x) = 3x 3 − x on the closed interval [0, 1] by the following steps:

i. Observe that f(x) is a polynomial, so it is continuous on the interval [0, 1].

ii. Compute the derivative f 0 (x), and show that it is equal to 0 at x = 1 3 and x = − 1 3 .

iii. Conclude that x = 1 3 is the only critical number in the interval [0, 1].

iv. Compute the values f(0), f(1), and f( 1 3 ); and indicate the maximum and minimum values of f(x) on [0, 1].

(b) Find the absolute maximum and minimum values of F(x) = x + 9 x on the closed interval [1, 20]. HINT: Use the same steps as the previous problem

Answer #1

Find the absolute maximum and absolute minimum values of f on
the given interval. f(x) = 3x^2 − 18x + 8, [0, 8] absolute minimum
value.

Find the absolute maximum and minimum values of f(x)=
−x^3−3x^2+4x+3, if any, over the interval
(−∞,+∞)(−∞,+∞).
I know it doesn't have absolute maxima and minima but where do
they occur? In other words x= ? for the maxima and minima?

Find the absolute maximum and minimum values of each function
over the indicated interval, and indicate the x-values at which
they occur.
f(x) = 3x^3 - 3x^2 - 3x + 7 ; [-1 , 0]
The absolute maximum value is _ at x = _
(Use a comma to separate answers as needed. Type an integer or
a fraction.)

1. The absolute maximum value of f(x) = x 3 − 3x 2 + 12 on the
interval [−2, 4] occurs at x =? Show your work.
2.t. Let f(x) = sin x + cos2 x. Find the absolute maximum, and
absolute minimum value of f on [0, π]. Show your work.
Absolute maximum:
Absolute minimum:
3.Let f(x) = x √ (x − 2). The critical numbers of f are_______.
Show your work.

Verify that the function
f(x)=(1/3)x3+x2−3x attains an absolute
maximum and absolute minimum on [0,2]. Find the absolute maximum
and minimum values for f(x) on [0,2].

Find the absolute maximum and minimum values on the closed
interval [-1,8] for the function below. If a maximum or minimum
value does not exist, enter NONE.
f(x) = 1 − x^2/3
Find the absolute maximum and absolute minimum values of the
function below. If an absolute maximum or minimum does not exist,
enter NONE.
f(x) = x3 - 12x on the
closed interval [-3,5]

find the absolute maximum and absolute minimum values of f on
the given closed interval
f(x)=5-x^2
[-3,1]

Let f(x) = x 1/2 (3−x). Find the absolute maximum and absolute
minimum values of the f(x) on the interval [1, 3].

Find the absolute maximum and minimum values of the following
function over the indicated interval, and indicate the x-values
at which they occur.
f(x)=1/3x^3+3/2x^2-4x+4 [-5,3]

Find the absolute maximum and minimum values of the function
over the indicated interval, and indicate the x-values at which
they occur.
f(x)=3x/x^2+9 ; [-8,8]
The absolute maximum value is___ at x=____

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