Welcome :: Homework Help and Answers :: Mathskey.com
Welcome to Mathskey.com Question & Answers Community. Ask any math/science homework question and receive answers from other members of the community.

13,435 questions

17,804 answers

1,438 comments

774,937 users

Verify that the function f satisfies the hypotheses of the Mean Value Theorem on the given interval.

0 votes
. Verify that the function f 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.

(a) f(x) = 7x − sin(2x), [− π /2 , π /2 ].

(b) f(x) = (2x−1)/( 3x+2) , [0, 2].
asked Mar 27, 2015 in CALCULUS by anonymous

2 Answers

0 votes

Step 1:

(a)

The function is .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on .

2. f is differentiable on .

Then there is a number c in such that  .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate  with respect to .

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

.

Step 4:

Substitute in .

image

General solution of is .

image

If then image.

If then image.

image are not in the interval , hence they are not considered.

Solution:

image.

answered Mar 27, 2015 by Lucy Mentor
0 votes

Step 1:

(b)

The function is .

Mean value theorem :

Let f be a function that satisfies the following three hypotheses :

1. f is continuous on

2. f is differentiable on

Then there is a number c in such that .

Step 2:

The function is .

The function is continuous on the interval .

Differentiate with respect to .

The function is differentiable on the interval .

Then .

Step 3:

From the mean value theorem :

image

.

Step 4:

Substitute in .

image

image are not in the interval , hence it is not considered.

Solution:

image.

answered Mar 27, 2015 by Lucy Mentor

Related questions

...