Question

1. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

f(x) = x^{3} + x − 5, [0, 2]

a) No, *f* is continuous on [0, 2] but not differentiable
on (0, 2).

b) Yes, it does not matter if *f* is continuous or
differentiable; every function satisfies the Mean Value
Theorem.

c) There is not enough information to verify if this function satisfies the Mean Value Theorem.

d) Yes, *f* is continuous on [0, 2] and differentiable on
(0, 2) since polynomials are continuous and differentiable on .

e) No, *f* is not continuous on [0, 2].

If it satisfies the hypotheses, find all numbers *c* that
satisfy the conclusion of the Mean Value Theorem. (Enter your
answers as a comma-separated list. If it does not satisfy the
hypotheses, enter DNE).

c = ??????????

2. If *f*(4) = 1 and *f '*(*x*) ≥ 1 for 4 ≤
*x* ≤ 9, how small can *f*(9) possibly be?

Answer #1

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 9, [0, 2]
Yes, f is continuous on [0, 2] and differentiable on
(0, 2) since polynomials are continuous and differentiable on .No,
f is not continuous on [0, 2]. No,
f is continuous on [0, 2] but not differentiable on (0,
2).Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.There is...

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval? f(x) = 3x2 + 3x + 6, [−1, 1]
No, f is continuous on [−1, 1] but not differentiable
on (−1, 1).
There is not enough information to verify if this function
satisfies the Mean Value Theorem.
Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
No, f is not continuous on [−1, 1].Yes, f is...

1aDoes the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = 4x2 + 3x + 1, [−1,
1]
a.No, f is continuous on [−1, 1] but not differentiable
on (−1, 1).
b.Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c.Yes, f is continuous on [−1, 1] and differentiable on
(−1, 1) since polynomials are continuous and differentiable on
.
d.No, f is not continuous on...

1) Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem. (Enter your answers as a
comma-separated list.)
f(x) = 1 − 12x + 2x^2, [2, 4]
c =
2) If f(2) = 7 and f '(x) ≥ 1 for 2 ≤
x ≤ 4, how small can f(4) possibly be?
3) Does the function satisfy the hypotheses of the Mean Value
Theorem...

Verify that the function satisfies the hypotheses of the Mean
Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f(x) = e^-x , [0,2]

Verify that the function satisfies the hypotheses of the mean
value theorem in the given interval. Then
find all the numbers x \ c that satisfy the conclusion of the mean
value theorem.
a. ?(?) = 2? 2 − 3? + 1,[0,2]
b. ?(?) = x 3 − 3x + 2,[−2,2]

Verify that the following function satisfies the hypotheses of
the Mean Value Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of the Mean Value Theorem.
f(x) = x3 - 3x + 1; [-2,2]
Step-by-step instructions would be very much appreciated. I'm a
little thrown off by the third power. Thank you for the help in
advance!

1. Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem. (Enter your answers as a
comma-separated list.)
f(x) = 7 − 24x + 2x^2, [5, 7]

Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of Rolle's Theorem. (Enter
your answers as a comma-separated list.)
f(x) = 1 − 24x +
4x2, [2, 4]

1) Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers
c that satisfy the conclusion of Rolle's Theorem. (Enter
your answers as a comma-separated list.)
f(x) = 7 − 8x +
2x2, [1, 3]

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