Question

Where is the Mean Value Theorem true?

f(x) = 3x - x^2 Find the x value on the interval [1, 4] where the conclusion of the Mean Value Theorem is true.

Answer 1

Given `f(x)=3x-x^2`, `x=5/2` according to the Mean Value Theorem.

Explanation:

First, check if f(x) is continuous, and since there are no restrictions, it is.
Second, check if it is differentiable, which it seems to be.
Lastly, since it is continuous and differentiable then `f'(x)=(f(1)-f(4))/(1-4)`
1. `f'(x)=3-2x`
2. `(f(1)-f(4))/(1-4)=(2-(-4))/(-3)=6/-3=-2`
3. Now set the two equal to each other.
`3-2x=-2`
`-2x=-5`
`x=5/2`