Justification with the mean value theorem: equation | AP Calculus AB | Khan Academy

Let g of x equal one over x. Can we use the mean value theorem to say that the equation g prime of x is equal to one half has a solution where negative one is less than x is less than two? If so, write a justification.

All right, pause this video and see if you can figure that out.

So the key to using the mean value theorem, even before you even think about using it, you have to make sure that you are continuous over the closed interval and differentiable over the open interval. So this is the open interval here, and then the closed interval would include the endpoints. But you might immediately realize that both of these intervals contain x equals 0, and at x equals 0 the function is undefined. And if it's undefined there, well, it's not going to be continuous or differentiable at that point.

And so no, not continuous or differentiable over the interval.

All right, let's do the second part. Can we use the mean value theorem to say that there is a value c such that g prime of c is equal to negative one half and one is less than c is less than two? If so, write a justification.

So pause the video again.

All right, so in this situation between 1 and 2 on both the open and the closed intervals, well, this is a rational function, and a rational function is going to be continuous and differentiable at every point in its domain. And its domain completely contains this open and closed interval. Or another way to think about it: every point on this open interval and on the closed interval is in the domain.

So we can write g of x is a rational function, which lets us know that it is continuous and differentiable at every point in this domain, at every point in its domain. The closed interval from 1 to 2 is in domain.

And so now let's see what the average rate of change is from 1 to 2.

And so we get g of two minus g of one over two minus one is equal to one half minus one over one, which is equal to negative one half.

Therefore, by the mean value theorem, there must be a c where one is less than c is less than two, and g prime of c is equal to the average rate of change between the endpoints, negative one half.

And we're done. So we could put a big yes right over there, and then this is our justification.