# Consider the following function. f ( x ) = 1 − x 2 / 3 Find f ( − 1 ) and f ( 1 ) . f ( − 1 ) = f ( 1 ) = Find all values c in ( − 1 , 1 ) such that f ' ( c ) = 0 . (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) c = Based off of this information, what conclusions can be made about Rolle's Theorem? This contradicts Rolle's Theorem, since f ( − 1 ) = f ( 1 ) , there should exist a number c in ( − 1 , 1 ) such that f ' ( c ) = 0 . This does not contradict Rolle's Theorem, since f ' ( 0 ) = 0 , and 0 is in the interval ( − 1 , 1 ) . This does not contradict Rolle's Theorem, since f ' ( 0 ) does not exist, and so f is not differentiable on ( − 1 , 1 ) . This contradicts Rolle's Theorem, since f is differentiable, f ( − 1 ) = f ( 1 ) , and f ' ( c ) = 0 exists, but c is not in ( − 1 , 1 ) . Nothing can be concluded.

(E)Nothing can be concluded.

Step-by-step explanation:

Given the function

If the derivative is set equal to zero, the function is undefined.

Nothing can be concluded since and no such c in (-1,1) exists such that

THEOREM

Rolle's theorem states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative is zero.

The function f(x) = 1 - x^2/3 has f(-1) = f(1) = 2/3. The derivative f'(x) = -2x/3 equals zero at x=0, which is in the interval (-1, 1). Therefore, this does not contradict Rolle's Theorem.

### Explanation:

The function given is f ( x ) = 1 - x ^ 2 /3. To find the values f(-1) and f(1), we simply substitute these values into the function. Therefore, f(-1) = 1 - (-1) ^ 2 /3 = 1 - 1/3 = 2/3 and f(1) = 1 - 1^2/3 = 2/3. As you can see, f(-1) = f(1).

Now, to find the value 'c' such that f'(c) = 0, first we need to determine the derivative of the function, f'(x) = -2x/3. Setting this equal to zero gives the equation 0 = -2x/3, which has the solution x = 0. Therefore, f'(c) = 0 at c = 0, which is within the interval (-1, 1).

Finally, regarding Rolle's Theorem which states that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the interval (a, b) such that f'(c) = 0, our results are consistent with Rolle's Theorem, since f is differentiable, f(-1) = f(1), and a 'c' value exists in the interval (-1, 1) such that f'(c) = 0.

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## Related Questions

2 3/4 divided by 4 1/8 in fraction form

Step-by-step explanation:

2 3/4 ÷ 4 1/8

23/4×8/41

23×8/4×41

46/41

=1.122

The solution after divide 2 3/4 by 4 1/8 in fraction form is,

⇒ 2/3

We have to given that,

To divide 2 3/4 by 4 1/8.

Simplify it as,

2 3/4 ÷ 4 1/8

11/4 ÷ 33/8

11/4 x 8/33

(11 x 8) / (4x33)

2/3

Therefore, The solution after divide 2 3/4 by 4 1/8 in fraction form is,

⇒ 2/3

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The mean of Greg’s journey times to work on 3 days was 28 minutes.If it took him 27 minutes and 25 minutes on two of the days, how long did it take him on the third day?

the mean is found by adding the time it takes Greg each day and then diving by the number of days: 27+25+32=84.
84/3=28 min
it takes Greg 32 min on the third day

28x3=84

84-27-25=32

That is my solution.

What is te average number of tomatoes on the 10 plants that were randomly selected ?

what were the options

Solve for c. a (c+b)=d

Step-by-step explanation:

a( c + b ) = d

Divide both sides by a

That's

Move b to the right side of the equation to make c stand alone

We have the final answer as

Hope this helps you

c = - b.

Step-by-step explanation:

a(c + b) = d

(c + b) = d / a

c + b =

c = - b.

Hope this helps!

X=1

3*4=12
x*4=4

Or

12/4=3
4/4=1

x=1

Step-by-step explanation:

i did it

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed. 51  53  48  62  34  34  51  53  48  30  62  51  46 ​(a) Find the sample mean and sample standard deviation. ​(b) Each employee in the sample is given a ​$5000 raise. Find the sample mean and sample standard deviation for the revised data set. ​(c) Each employee in the sample takes a pay cut of ​$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. ​(d) What can you conclude from the results of​ (a), (b), and​ (c)?

Mean increase or decrease (same quantity) according to the quantity of the increment or reduction

As all elements were equally affected the standard deviation will remain the same

Step-by-step explanation:

For the original set of salaries: ( In thousands of $) 51, 53, 48, 62, 34, 34, 51, 53, 48, 30, 62, 51, 46 Mean = μ₀ = 47,92 Standard deviation = σ = 9,56 If we raise all salaries in the same amount ( 5 000$ ), the nw set becomes

56,58,53,67,39,39,56,58,53,35,67,56,51

Mean   =  μ₀´  = 52,92

Standard deviation  =  σ´ = 9,56

And if we reduce salaries in the same quantity ( 2000 $) the set is 49,51,46,60,32,32,49,51,46,28,60,49,44 Mean μ₀´´ = 45,92 Standard deviation σ´´ = 9,56 What we observe 1.-The uniform increase of salaries, increase the mean in the same amount 2.-The uniform reduction of salaries, reduce the mean in the same quantity 3.-The standard deviation in all the sets remains the same. We can describe the situation as a translation of the set along x-axis (salaries). If we normalized the three curves we will get a taller curve (in the first case) and a smaller one in the second, but the data spread around the mean will be the same Any uniform change in the data will directly affect the mean value Uniform changes in values in data set will keep standard deviation constant ### Final answer: The mean salary is affected by each employee's changes in salary, such as raises and pay cuts, but the standard deviation (the spread of salaries) remains the same provided the change is the same for all individuals. ### Explanation: To answer this question, we need to calculate the sample mean and sample standard deviation in each case. The sample mean is the average of the data, while the sample standard deviation is a measure of the amount of variation or dispersion in the data set. • (a) Calculate the sample mean and sample standard deviation of the initial salaries. This involves summing all the salaries and dividing by the total number to get the mean, then calculating the standard deviation using the formula: square root of [sum of (each salary - mean salary)² divided by (total number of salaries - 1)]. • (b) When each employee is given a$5000 raise, the mean will increase by 5, while the standard deviation will remain the same because raises do not affect the dispersion of the team's salaries.
• (c) Similarly, when each employee takes a pay cut of \$2000, the mean salary will decrease by 2, but the standard deviation will again remain the same because pay cuts do not affect the dispersion of the team's salaries.
• (d) We can conclude that while the mean is affected by changes in individual salaries (like raises and pay cuts), the standard deviation is not, provided the change is the same for all individuals. Therefore, it shows that while supply changes can affect the central tendency (mean), they do not impact how spread out the salaries are (standard deviation).