Provide a sketch of a function that exhibits the hypotheses and conclusion of the Mean Value theorem.

Question

State the Mean Value Theorem. Provide a sketch of a function that exhibits the hypotheses and conclusion of the Mean Value theorem.

Answer 1

Step 1:

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:

Let us consider a function that exhibits the hypotheses.

The function is .

The function is continuous on the interval .

Differentiate  with respect to .

The function is differentiable on the interval .

Then .

Answer 2

Contd....

Step 3:

From the mean value theorem :

.

Step 4:

Substitute in .

General solution of is .

If then .

If then .

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

Therefore, .

Step 5:

Sketch the function to show graphically that the function that exhibits the hypotheses.

Therefore function exhibits the hypotheses.

Solution:

Any function that satisfies the following three hypotheses :

1. f is continuous on .

2. f is differentiable on .

Will have a number c in .