1. ## Question about a function

Assuming the conditions for the MVT hold for $f:\left[ {a,a + h} \right] \to$R, so that for some $\theta \in \left( {0,1} \right)$ we have

$f\left( {a + h} \right) - f\left( a \right) = hf'\left( {a + \theta h} \right)$.

If we fix f and a for each non-zero h, how would you write $\theta \left( h \right)$ for a corresponding value of theta?

2. Originally Posted by james89
Assuming the conditions for the MVT hold for $f:\left[ {a,a + h} \right] \to$R, so that for some $\theta \in \left( {0,1} \right)$ we have

$f\left( {a + h} \right) - f\left( a \right) = hf'\left( {a + \theta h} \right)$.

If we fix f and a for each non-zero h, how would you write $\theta \left( h \right)$ for a corresponding value of theta?
You can't. All the MVT tells you is that f(a+h)- f(z)= h f( $\xi$) but says nothing about where $\xi$ is.

3. Originally Posted by HallsofIvy
You can't. All the MVT tells you is that f(a+h)- f(z)= h f( $^xi$) but says nothing about where $\xi$ is.
It is definitely possible to complete with the information given - otherwise I wouldn't have been asked it.

4. Originally Posted by james89
It is definitely possible to complete with the information given - otherwise I wouldn't have been asked it.
Maybe you should explain your question more clearly.

5. I'm not sure what else I can say beyond what I have already given. The 'R' in the function definition is the set of real numbers, and I have been asked "Fix f and a, and for each non-zero h write theta(h) for a corresponding value of theta."

6. Originally Posted by james89
I'm not sure what else I can say beyond what I have already given. The 'R' in the function definition is the set of real numbers, and I have been asked "Fix f and a, and for each non-zero h write theta(h) for a corresponding value of theta."
In that case, I must agree with Halls.
You said that you know it is possible. How do you know that?
It may be that whoever wrote the question has a complete answer in mind.
If I were you, I would discuss this the source of the question.