Question

Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers  that satisfy the onclusion of Rolle’s Theorem.

f (x) = √x - 1/3 x,        [0, 9]

Answer 1

Step 1 :

Rolle's Theorem :

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

1. f  is continuous on .

2. f  is differentiable on .

3. .

Then there is a number c in such that .

Step 2 :

The function is and the interval .

Since the function is a polynomial, it is continuous on the interval .

Differentiate with respect to x.

is not differentiable at , but it is differentiable on the open interval , which satisfies the Rolle's Theorem.

Step 3 :

Check at the end points :

At :

At :

, hence Rolls theorem is applicable on .

Step 4 :

Find the value of c, such that .

The number .

Solution :

The number .