Question

Explain why Rolle's Theorem does not apply to the function even though there exist a and b such that  f (a) = f (b).

f (x) = |1/x|,            [-1, 1]

Answer 1

Step 1 :

Rolle's Theorem :

Let be a function that satisfies the following three hypotheses.

1.   is continuous on .

2.   is differentiable on .

3. .

Then there is a number   in such that .

Step 2 :

.

Step 3 :

The function is and the interval is  .

A function is continuous when, for every value in its domain, is defined, and .

Consider the number as .

Since the function is undefined at , it is discontinuous on the interval .

Since the function is discontinuous on the interval , it does not satisfies the Rolle's Theorem.

Solution :

.

Since the function is discontinuous on the interval , it does not satisfies the Rolle's Theorem.