Calculus > Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a,b] and if the function is differentiable on the open interval (a,b) then there is a number c in (a,b) such that:

–which can be read as “the first derivative at c is equal to the function value at b, minus the function value at a, divided by b-a. **What does this mean in English? **The theorem can be applied to many real-life situations. For example, if you are driving a car, you could find your average speed, f'(c), by dividing the distance you traveled (f(b)-f(a)) by the time you took (b-a). The Mean Value Theorem states that there must be at least one moment when your speed was equal to your average speed.

## Mean Value Theorem Sample Problem

**Sample problem:** Find a value of c for f(x) = 1 + ^{3}√(x-1) on the interval [2,9] that satisfies the mean value theorem.

Step 1: Check that the function is continuous and differentiable. This particular function — a cubed root — is both differentiable and continuous.

Step 2: **Find the derivative**. Use the chain rule for this particular function.

Step 3: **Plug the derivative into the left side of the formula**.

Step 4: Plug the function **inputs **(from the question) and the **function’s values** into the right side of the mean value theorem formula.

Step 5: **Work the right side of the equation**, evaluating the function (from the question) at f(2) and f(9).

Step 6: **Solve for x, using algebra:**

- Multiplying both sides by 3, rewriting as
^{2}⁄_{3}power.

- 2. Multiplying by (x-1)
^{2}⁄_{3}

- Multiplying both sides by
^{7}⁄_{3}

- 4. Raising both sides to
^{3}⁄_{2}

- Adding 1 to both sides

A negative can’t be part of the solution (considering that the interval [2,9] is positive) so the point that solves the equation is:

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