# Lagrange's mean value theorem problem

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1. Jan 31, 2015

### utkarsh009

1. The problem statement, all variables and given/known data

2. Relevant equations

Lagrange's mean value theorem
3. The attempt at a solution
Applying LMVT,
There exists c belonging to (0,1) which satisfies f'(c) = f(1)-f(0)/1 = -f(0)
But this gets me nowhere close to the options... :(

2. Jan 31, 2015

### O_o

edit
Sorry the post I made contained an error. I'm not quite sure how to fix it yet but basically what I think you need to do is find a function h(x) and define it in terms of some linear combination of f(x), f(0), f(1) so that it equals zero at the end points and then apply Rolle's Theorem.

3. Jan 31, 2015

### utkarsh009

Thanks mate!! Solved it... But how did you think of that function h(x)??

4. Jan 31, 2015

### utkarsh009

5. Jan 31, 2015

### O_o

Hey sorry I edited my previous post. What I wrote doesn't work because the two c's might be different. The c that works for the equation you posted might be a different c from the one I used for h(x).

When trying to define an h(x) we want it to be 0 at the end points so we can apply Rolle's Theorem. It looks like Rolle's Theorem is needed because of the way the question is set up (i.e it looks like some derivative is equal to 0). And we want f(0) to disappear from the equation for the derivative because we don't know its value.

6. Jan 31, 2015

### O_o

Actually a relatively simple function works lol. Sorry for all this confusion:
$$h(x) = xf(x)$$
There exists a c in (0, 1) such that $$0 = cf'(c) + f(c)$$

by Rolle's Theorem.

7. Jan 31, 2015

### utkarsh009

Yah... Was just thinking the same and was about to post it... :P