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.
Answers
Answer:
(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.
Final answer:
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.
Learn more about Rolle's Theorem here:
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Related Questions
Someone please give me all the answers!
2x + 3y = 12Complete the missing value in the solution to the equation.,8)
What is the sum in simplest form?4 1/2 + 1 3/5
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Answer:
P2 affirms P1 and the conclusion is in the same direction.
P1--->P2--->C
This argument is valid.
Step-by-step explanation: using the syllogism rules.
Premises 1 (P1) = Some foreign emissaries are persons without diplomatic immunity,
Premises 2 (P2) = so some persons invulnerable to arrest and prosecution are foreign emissaries
Conclusion (C) = because no persons with diplomatic immunity are persons vulnerable to arrest and prosecution.
From the argument:
P1 uses "some", that means it's not "all" foreign emissaries person that does not have diplomatic immunity. This means that some other foreign emissaries have diplomatic immunity
P2 uses "some", that means it's affirms to that part of P1 which states that some foreign emissaries have diplomatic immunity.
The conclusion is valid because the part of P2 which states that some foreign emissaries are vulnerable to arrest, which affirms with P1 which states that Some foreign emissaries are persons without diplomatic immunity. That means no persons with diplomatic immunity are persons vulnerable to arrest and prosecution. This conclusion literally means that if you don't have diplomatic immunity, you are vulnerable to arrest and prosecution.
Therefore;
P2 affirms P1 and the conclusion is in the same direction.
P1--->P2--->C
This argument is valid.
Answers
Answers
The ratio in its simplest form is 1/4.
We need to determine the ratio in the simplest form between 30 minutes and 2 hours.
To obtain the required ratio, the calculation must be done in a single unit of measurements, therefore change the hour calculation into minutes.
We know that 1 hour is equivalent to 60 minutes.
Therefore,
Thus, the required ratio is calculated below:
Hence, the ratio in its simplest form is 1/4.
For more information about Ratio click the link given below.
experiment does not led
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Answer:
D: the wiring and the other
Step-by-step explanation:
Answer:
The liquid and the medal
Step-by-step explanation: "So the electrons start to move from one metal to the other through the liquid. This creates current" This is why i would check the liquid and the medal. if the experiemnt doesnt work then there must be something wrong with the current
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Answer:
5
Step-by-step explanation:
Answer:
Step-by-step explanation:
You have to use PEMDAS
first p, parenthesis
12-2=10
then e exponenets 3^3=9
and finally d diviison /2
so 9(10) : 2
=90/2
=45
b. 2
c. -2x
d. -2
please explain
Answers
Answer:
-2
Step-by-step explanation:
y= Mx+b M= slope
y=-2x+b. -2 is in the m spot