Extreme Value Theorem | Mean Value Theorem |

Rolle’s Theorem | More Practice |

**Curve Sketching** isn’t my favorite subject in Calculus, since it’s so abstract, but it’s useful to be able to look at functions and their characteristics by simply taking derivatives and thinking about the functions. Before we get into curve sketching, let’s talk about two theorems that seems sort of obvious, but we need to go over them nonetheless.

To sketch curves in Calculus, we’ll be looking at **minimums** and **maximums** of functions in certain intervals, so we have to talk about a few theorems first. We’ll need these theorems to know that if a function is differentiable and the **derivative at a certain point is 0**, then that point is either a **minimum **or **maximum.** Thus, before you to get to actual curve sketching, you’ll probably see some problems as in this section.

## Extreme Value Theorem

The **Extreme Value Theorem** states that **a function on a closed interval must have both a minimum and maximum in that interval**. (A closed interval is an interval that includes its endpoints, which are the points at the very beginning and end of the interval). If we didn’t include the endpoints, we’d have an open interval, and we may never have a minimum or maximum, since the function could get closer and closer to a point but never touch it (like we saw with **Limits**). When we do include these endpoints, there will definitely be a minimum or maximum; for example, if the function is increasing for the whole interval, the minimum and maximum would be at these endpoints.

## Rolle’s Theorem

**Rolle’s Theorem** states that if the function in an interval comes up and back down (or down and back up) and ends up exactly where it started, you’ll have at **least one maximum or minimum** (where the **derivative is 0**). Here is the formal form of Rolle’s Theorem:

**Rolle’s** **Theorem:**

If a function is continuous on a closed interval $ [a,b]$ and differentiable on the open interval $ (a,b)$, and $ f\left( a \right)=f\left( b \right)$ (the $ y$’s on the endpoints are the same), then there is at least one number $ c$ in $ (a,b)$, where $ {f}’\left( c \right)=0$.

Note that if this function was not differentiable, you would still have either a maximum or minimum, but you may not be able to take the derivative at that point; for example, you may have a sharp turn instead of a nice curve at that point.

Here are pictures of a differentiable and non-differentiable functions. For the differentiable graph, do you see how if the graph goes up and comes back down, we have to have at least one point where the derivative is **0** (at the maximum)?

## Mean Value Theorem

Note that the **Mean Value Theorem for Integrals** can be found here in the **Definite Integration** section.

The **Mean Value Theorem** **for Derivatives** is a little bit more important and is proven using the Rolle’s theorem. It says that somewhere inside a closed interval $ [a,b]$ there exists a point $ c$ where the **derivative** at this point is the same as the **slope** between points $ a$ and $ b$. Think of the Mean Value Theorem as Rolle’s Theorem, but possibly “tilted”.

Here’s the formal form of the **Mean Value Theorem** and a picture; in this example, the **slope of the secant line is 1**, and also the **derivative at the point **$ \boldsymbol {(2,3)}$** (tangent line) is also 1**. We’ll see more examples below.

**Mean Value Theorem**

If a function is continuous on a closed interval $ \left[ {a,b} \right]$ and differentiable on the open interval $ \left( {a,b} \right)$, then there is at least one number $ c$ in $ \left( {a,b} \right)$, where $ \displaystyle {f}’\left( c \right)=\frac{{f\left( a \right)-f\left( b \right)}}{{b-a}}$.

Here are some problems that you might see with these theorems:

Here are a few more typical **Mean Value Theorem** (**MVT**) problems. **Note that when we get our value of **$ \boldsymbol {c}$**, we have to make sure it lies in the interval we’re given. **Note also that these problems may be worded something like this: For what value of $ c$ on a certain open interval would the **tangent** to the graph of a certain function be **parallel to the secant line** in that closed interval?

Here’s one more where we need to understand visually the **Mean Value Theorem:**

**Learn these rules, and practice, practice, practice!**

Click on **Submit** (the blue arrow to the right of the problem) and click on **Find Where the Mean Value Theorem is Satisfied** to see the answer. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Curve Sketching**** **– you are ready!