How do you use the mean value theorem to find roots?

Question

Scarlett Allison · Accepted Answer

I don't believe the Mean Value Theorem can be used to find roots.  The mean value theorem tells us that under certain conditions,
 (namely, #f# continuous on #[a,b]# and differentiable on #(a,b)#)
 there is a value in a specific interval (namely #(a,b)# ) that solves a certain equation. 
 (#f'(x)=(f(b)-f (a))/(b-a)#) The theorem gives no information on how to find a solution. It only asserts that a solution to a particular equation exists.

Axel Alvarez · Answer

undefined To use the Mean Value Theorem to find roots, follow these steps:

1. Identify the function $ f(x) $ for which you want to find the roots.
2. Choose an interval $[a, b]$ such that $ f(a) $ and $ f(b) $ have opposite signs.
3. Apply the Mean Value Theorem, which states that if a function $ f $ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists a point $ c $ in $(a, b)$ such that $ f'(c) = \frac{f(b) - f(a)}{b - a} $.
4. Set $ f(c) = 0 $ and solve for $ c $, which represents a root of the function $ f(x) $ within the interval $[a, b]$.

This method helps find a point within the given interval where the function crosses the x-axis, indicating a root.

Adam Adkins · Answer

undefined The Mean Value Theorem states that if a function $ f(x) $ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one number $ c $ in the interval $(a, b)$ such that $ f'(c) = \frac{{f(b) - f(a)}}{{b - a}} $.

To use the Mean Value Theorem to find roots, follow these steps:

1. Choose a function $ f(x) $ with the desired root.
2. Determine the interval $[a, b]$ where you suspect the root might lie.
3. Evaluate $ f(a) $ and $ f(b) $ to find the values of the function at the endpoints of the interval.
4. Calculate the average rate of change of the function over the interval using the formula $ \frac{{f(b) - f(a)}}{{b - a}} $.
5. Find $ c $ such that $ f'(c) $ equals the average rate of change calculated in step 4.
6. If $ f(c) = 0 $, then $ c $ is a root of the function.

This process relies on the Intermediate Value Theorem, which guarantees the existence of a root for a continuous function that changes sign over an interval. By applying the Mean Value Theorem, we can narrow down the location of the root within that interval.