Question

Is the mean value theorem can be applied to `f(x)= 1/x` in the interval `[-1, 1]`?

Can't be applied

`f(x)= 1/x`
`f'(x) = -1/x^2`
`f(c)= -1/c^2`

Applying Mean Value Theorem:
`f'(c)=(f(b)-f(a))/(b-a)`
`=(1-(-1))/(1-(-1))=1`

For any value of `c` there will be no value of `f'(c) = 1`

Therefore, Mean value theorem can't be applied.

Please verify my answer. How can I describe it in a better way?

Answer 1

Please see below.

Explanation:

When we are asked whether some conditional theorem "can be applied" to some situation, we are usually really being asked "Can we use the theorem to show that the conclusion is true?"

We can do this if the hypotheses are true. So the question amounts to "Are the hypotheses of this theorem true in this situation?"

To answer this particular question, we need to determine whether the hypotheses of the Mean Value Theorem are true for the function

`f(x) = 1/x` on the interval `[-1,1]`

The Mean Value Theorem has two hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`
H2 : `f` is differentiable on the open interval `(a,b)`.

We say that we can apply the Mean Vaue Theorem if both hypotheses are true.

H1 : The function `f` in this problem is not continuous on `[-1,1]` [Because, this function is undefined at `0` which is in the interval.]

That is actually enough to tell us the the theorem "cannot be applied".

Let's check the other hypothesis anyway.

H2 : The function `f` in this problem is not differentiable on the entire interval `(-1,1)`
[Because the derivative, `f'(x) = -1/x^2` fails to exist at `0` which is in the interval `(-1,1)`.)]

A note on "if . . . then . . . " theorems

if the hypotheses are not true, we learn nothing about the truth of the conclusion.

For an example involving the Mean Value Theorem, see this question and answer:

https://socratic.org/questions/how-do-you-determine-whether-the-function-satisfies-the-hypotheses-of-the-mean-v