How do you give a value of c that satisfies the conclusion of the Mean Value Theorem for Derivatives for the function f(x)=-2x^2-x+2 on the interval [1,3]?

1 Answer
Jim H
Mar 23, 2015

The conclusion of the Mean Value Theorem says that there is a number c in the interval (1, 3) such that:
f'(c)=(f(3)-f(1))/(3-1)

To find (or try to find) c, set up this equation and solve for c.
If there's more than one c make sure you get the one (or more) in the interval (1, 3).

For f(x)--2x^2-x+2, we have
f(1)=-1, and f(3)=-18-3+2=-19
Also,
f'(x)=-4x-1.

So the c we're looking for satisfies:

f'(c)=-4c-1=(f(3)-f(1))/(3-1)=(-19--1)/(3-1)=(-18)/2=-9

So we need

-4c-1=-9. And c=2.

Note:
I hope you've been told that actually finding the value of c is not a part of the Mean Value Theorem.
The additional question"find the value of c" is intended as a review of your ability to solve equations. For most functions, you will not be able to find the c that the MVT guarantees us is there..

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