How To Use Rolle's Theorem

What is Rolle's Theorem?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching rolle's theorem

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

And how is it useful?

All good questions that'll be explained shortly in today's lesson.

Let's go!

Imagine you're a detective and you are tracking a suspect. You know the precise moment the suspect left their house, and too when they returned home again, but you are curious as to what they did while abroad. What path did they take? Did they make whatsoever foreign turns or stops along the style?

Oddly enough, Rolle'due south Theorem can help us assemble the clues.

How?

Rolle'due south theorem has a way of guaranteeing that there is a to the lowest degree 1 betoken (moment) within a closed interval (leaving and returning to the house) at which the derivative is zippo (suspect makes a turn along their path).

Let's larn how.

Defined w/ A Stride-past-Stride Procedure

Rolle's Theorem states that if f is a continuous function on the closed interval [a,b], differentiable in the open interval (a,b), and f(a)=f(b) then there exists at least one number c in (a,b) such that the f'(c) = 0.

But what does this theorem really mean?

Let's use our suspicious suspect to sort this out. Let a and b be our time, and f(a) and f(b) be the suspect'south house. And then, the suspect leaves the house f(a) at time a and returns onetime later. This would be time t=b and since the doubtable is returning to the same identify they left, we know that f(a) is equal to f(b).

At present, if nosotros can bear witness that our doubtable stays on a continuous path while they are away from the house, so at that place will come a moment when the suspect must end and plough (i.east., derivative equal to zero) in order to return dwelling house once again.

And that's all their is to Rolle's Theorem. If nosotros are given a closed interval where the y-values equal, and the path is continuous, so at some signal inside the interval the slope will be aught.

Rolle'southward Theorem is a unproblematic 3-pace procedure:

Check to make certain the role is continuous and differentiable on the airtight interval.
Plug in both endpoints into the office to check they yield the same y-value.
If aye, to both steps above, then this means we are guaranteed at least ane point within the interval where the first derivative equals nix.

Worked Example

Let'south expect at an case to run into this in activeness.

Suppose we are asked to determine whether Rolle's theorem can be applied to \(f(x)=x^{four}-2 x^{2}\) on the closed interval [-2,2]. And if and so, find all values of c in the interval that satisfy the theorem's conclusion.

Step 1:

Okay, then first, we will check to see that f(x) is a continuous and differentiable role on the interval.
Since \(f(ten)=10^{4}-ii 10^{2}\) is a polynomial role, then f(x) is continuous and differentiable.

Stride two:

Now, we must verify that the y-values at the endpoints are the same.

\begin{equation}
\begin{assortment}{l}
f(-2)=(-2)^{4}-2(-two)^{ii}=8 \\
f(2)=(ii)^{4}-two(2)^{ii}=8
\end{array}
\finish{equation}

Pace 3:

Considering they both yield the same y-value of 8, we know that all requirements are satisfied, which means we can at present find all values of c within the open interval (-2,ii) where f'(10) = 0.

\begin{equation}
\brainstorm{array}{l}
f^{\prime}(10)=4 ten^{3}-four x \\
4 x^{three}-4 ten=0 \\
4 x\left(x^{two}-one\right)=0 \\
4 x(x-i)(x+ane)=0 \\
x=0,one,-1
\cease{assortment}
\cease{equation}

Therefore, by using this process, we have establish iii values where the slope of the tangent line is zero within the interval!

rolles theorem closed interval max min

Rolles Theorem Closed Interval Max Min

Meet how easy that is!

Rolle'due south Theorem Vs Extreme Value Theorem

It's important to note the subtle simply important difference between Rolle's Theorem and the Extreme Value Theorem.

Every bit a reminder, the Extreme Value Theorem states that a continuous part on a closed interval [a,b] must have both a minimum and maximum on the interval. These extrema values tin can, and often do, occur at the endpoints.

absolute maximum and minimum

Accented Maximum And Minimum

Just what Rolle's Theorem guarantees are the being of an extreme value in the open interval, not including the endpoints.

rolles theorem critical numbers

Rolle's Theorem – Critical Numbers

Together we will work through numerous examples, where we will starting time determine if Rolle'south Theorem applies and if so, nosotros volition discover the point(south) where.

Let'south become later on it!