Question

verify Rolle's theorems​

Answer 1

Answer:

If f is continuous on [a,b] and differentiable on (a,b), with f(a)=f(b), then there exists c∈(a,b) such that f(c)=0.

Step-by-step explanation:

The given function is f(x)=|9−x2| on [−3,3]; also find point where the derivative is zero.

I got the value of c as ±3, but according to the theorem the value should be in between −3 and 3, i.e., excluding the −3 and 3

The function |9−x2| is not differentiable on −3 and 3, so the derivatives cannot be zero at that point, as they do not exist at all. You must have made a mistake somewhere.

In your problem particularly, as f(x)=9−x2 we have f′(x)=−2x, as 9−x2≥0 on [−3,3], the point where the derivative is 0 is 0, and only 0.

Also, note that Rolle's Theorem says that the point where the derivatives are zero are all in [a,b] where f(a)=f(b). It merely says that there exists such a point.

For example, take f(x)=0 on [1,3]. There exists a value where f′(x)=0 on (1,3), but f′(1)=f′(3)=0 as well. So values where f′(x)=0 need not always exist in the interval (a,b).