1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
V125BC [204]
3 years ago
9

Consider the following function. f(x) = 16 − x2/3 Find f(−64) and f(64). f(−64) = f(64) = Find all values c in (−64, 64) such th

at 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 is differentiable, f(−64) = f(64), and f '(c) = 0 exists, but c is not in (−64, 64). This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−64, 64). This contradicts Rolle's Theorem, since f(−64) = f(64), there should exist a number c in (−64, 64) such that f '(c) = 0. This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−64, 64). Nothing can be concluded.
Mathematics
2 answers:
VARVARA [1.3K]3 years ago
6 0

Answer:

This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−64, 64).

Step-by-step explanation:

The given function is

f(x)=16-\frac{x^2}{3}

To find f(-64), we substitute x=-64 into the function.

f(-64)=16-\frac{(-64)^2}{3}

f(-64)=16-\frac{4096}{3}

f(-64)=-\frac{4048}{3}

To find f(64), we substitute x=64 into the function.

f(64)=16-\frac{(64)^2}{3}

f(64)=16-\frac{4096}{3}

f(64)=-\frac{4048}{3}

To find f'(c), we must first find f'(x).

f'(x)=-\frac{2x}{3}

This implies that;

f'(c)=-\frac{2c}{3}

f'(c)=0

\Rightarrow -\frac{2c}{3}=0

\Rightarrow -\frac{2c}{3}\times -\frac{3}{2}=0\times -\frac{3}{2}

c=0

For this function to satisfy the Rolle's Theorem;

It must be continuous on [-64,64].

It must be differentiable  on (-64,64).

and

f(-64)=f(64).

All the hypotheses are met, hence this does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−64, 64) is the correct choice.

oksian1 [2.3K]3 years ago
3 0

Answer:

c = DNE

Option C: This contradicts Rolle's Theorem, since f(−64) = f(64), there should exist a number c in (−64, 64) such that f '(c) = 0.

Step-by-step explanation:

The given function is

f(x)=16-x^{\frac{2}{3}}

Rolle's theorem states that if the function f is

1. Continuous on [a, b],

2. Differentiable on the open interval (a, b) such that f(a) = f(b),

then f′(x) = 0 for some x with a ≤ x ≤ b.

At x=64,

f(64)=16-(64)^{\frac{2}{3}}=0

At x=-64,

f(-64)=16-(-64)^{\frac{2}{3}}=0

So, f(−64) = f(64).

Differentiate the given function with respect to x.

f'(x)=0-\frac{2}{3}x^{-\frac{1}{3}}

f'(x)=-\frac{2}{3x^{\frac{1}{3}}}

Substitute x=c,

f'(c)=-\frac{2}{3c^{\frac{1}{3}}}

We need to find the value of c such that f '(c) = 0.

f'(c)=0

-\frac{2}{3c^{\frac{1}{3}}}=0

-2=0

This equation is not true for any value of c. So, the value of does not exist.

This contradicts Rolle's Theorem, since f(−64) = f(64), there should exist a number c in (−64, 64) such that f '(c) = 0.

Therefore, the correct option is C.

You might be interested in
Help please! I need to write 4y =-3x + 12 in standard form, find the y-intercept of the previous equation, as an ordered pair, a
damaskus [11]

Answer:

-3x+4y=12\\y-intercept: (0,3)\\x-intercept: (-4,0)

Step-by-step explanation:

The standard form of a linear equation is ax+by=c, where a , b, and c are constants in the equation.

Based on the equation you have provided, the standard form would be -3x+4y=12, where a=-3, b=4, c=12.

To find the y-intercept, set x=0.

-3(0)+4y=12\\0+4y=12\\4y=12\\y=\frac{12}{4} =3

Therefore, the y-intercept is (0,3).

Do the same thing for the x-intercept, but this time with y=0.

-3x+4(0)=12\\-3x+0=12\\-3x=12\\x=\frac{12}{-3} =-4

Therefore, the x-intercept is (-4,0).

5 0
3 years ago
an online retailer sells two packages of protein bars. (10 pack of 2.1 ounce bars for $15.37) (12 pack of 1.4 ounce bars for $15
Anna35 [415]
It is better if you purchase the 12 pack of 1.4 to point one ounce bars for 2 cents less  and get more and pay less and get more
6 0
3 years ago
Help please!:( <br><br> right answers only please
bagirrra123 [75]

Answer:

100 dollars a month I think

3 0
3 years ago
Read 2 more answers
You are at a restaurant and the check comes to a total of $20. If you want to leave a 20% tip, how much total money should you p
grandymaker [24]

Answer:

the total tip should be $24.00

7 0
3 years ago
if a pharmacy technician makes as error resulting in serious consequences who can be held liable in court for error of negligenc
beks73 [17]

The very pharmacy technician must be held guilty


3 0
3 years ago
Other questions:
  • A straight road rises at an inclination of 0.3 radian from horizontal. Find the slope and change in elevation over a one mile se
    8·1 answer
  • Find the percent of change<br> 48 notebooks to 14 notebooks
    9·1 answer
  • Can you construct a triangle that has side lengths 5 cm, 5cm, and 10 cm
    10·2 answers
  • A school has 17 tables in the cafeteria. Each table seats 12 students. What is the
    14·2 answers
  • Suppose the amount of exports of textile machinery from Italy to the rest of the world equals 60 billion tons. The amount of imp
    7·1 answer
  • Write 5.48 as a mixed number in simplest form.<br> 5.48=
    10·1 answer
  • Latrell is going to put chocolate ice cream in x bowls for his friends. He will put 0.75 cups of ice cream in each bowl. Which e
    15·1 answer
  • I need help i dont understand it. its due in literally 2 hrs.
    5·1 answer
  • Given F(x) = 7(1 - x), what is the value of F(-8) ?<br> Please help
    8·2 answers
  • Explain whether the following given sets are closed under addition:
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!