Determine all intervals which contain a number c satisfying h''(c) = 2

Question

consider a function h that is twice differentiable. This means that h is differentiable and its derivative,h' , is also differentiable. Some values of h'(x) are given in the table below

x -8 -6 -4 -2 0 2 4 6 8

h’(x) 3 7 0 -3 -5 -4 0 -2 6

Answer 1

I'd recommend leveraging a theorem called the Mean Value Theorem. You already know that h is twice differentiable and differentiability implies continuity. The Mean Value Theorem is phrased like this:

Given an interval [a, b] for a function f(x) that's differentiable over open interval (a, b) and continuous on the interval [a, b], there exists a value c in the interval [a, b] such that f'(c) = (f(b) - f(a)) / (b-a)

How on earth do we use this fact for this problem????

I would recommend identifying for each of the adjacent values in the table shown above the value of the average rate of change. For example, between x values in [-8, -6] in h'(x), the average rate of change is

(7 - 3) / (-6 -(-8)) = 4/2 = 2. We see that by the Mean Value Theorem, there is a value c in [-8, -6] where h''(c) = 2. I'd repeat this check for the following intervals for the x values:

* [-6, -4]

* [-4, -2]

* [-2, 0]

....

Hope this helps! Feel free to followup if you have any further doubts!