# How do you use Rolle's Theorem on a given function #f(x)#, assuming that #f(x)# is not a polynomial?

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I know how to apply the theorem for polynomials, but I am unsure on how to apply it for other types of function.

I know how to apply the theorem for polynomials, but I am unsure on how to apply it for other types of function.

##### 2 Answers

See below.

#### Explanation:

To use Rolle's Theorem on any kind of function we need

a function,

We determine whether the function is continuous on the closed interval

We also determine whether the function is differentiable (has a derivative) on the open interval

We need to determine whether

If all three conditions are met, then we may cite Rolle's Theorem to conclude that there is a

The only real differences between applying Rolle's to polynomial and non-polynomial functions are

the function could fail to be continuous on

#[a,b]# or fail to be differentiable on #(a,b),it will take more to explain why we think the function does or does not satisfy the conditions for conituity and differentiabiity.

I'm not sure that this will fully answer your question, but I con't explain more without some more questions from you.

Here are some examples.

#### Explanation:

It is not too difficult to see that, for any

This is a rational function, so it is continuous at every number for which it is defined.

The function is not defined, hence not continuous, at

(Because differentiabillity implies continuity, the function is also not differentiable at

**For example:**

Therefore, by Rolle's Theorem, there is a

**Second example (same function)**

That is enough to tell us that we cannot use Rolle's Theorem for this function on this interval.

For the sake of the example, I will point out that

We do have

NOTE THAT: although we cannot cite Rolle's Theorem on tis interval, it is still true that there is a

**Third example (different function)**

(

Although

In fact, since