Rolle's Theorem Problem with f(a)=f(b)=0: Finding f'(c)=f(c)/c on [a,b]

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In summary, the conversation discusses a problem that appears to be a Rolle's theorem problem, but the solution is not working out. The problem involves finding the value of c between a and b where f'(c)=f(c)/c, given that f is continuous on [a,b] and differentiable on (a,b) with f(a) = f(b) = 0. The person has spent 3 hours trying to solve it and is seeking advice from someone who knows the answer. Eventually, they realize they can use Rolle's theorem on g(x) = f(x)/x and thank the person for their help.
  • #1
xXPhoenixFireXx
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Anyway, I've come across this problem that I can't figure out. It looks set up as a Rolle's Thm problem, but it just doesn't work out...

Let f be a continuous function on [a,b] and differentiable on (a,b) for some a,b > 0. Suppose f(a) = f(b) = 0.

Show that f'(c)=f(c)/c for some c between a and b.

The thing is this is next to straightforward problems like the integral of xsinx and |3-2x|>1. Am I just missing something?

I mean, I've spent about 3 hours looking for a similar problem/theorem in my Calc book; even just a pointer from someone who knows the answer would be great.
 
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  • #2
Use Rolle's theorem on g(x) = f(x)/x
 
  • #3
*smacks self hard*

I can't believe I missed that, thanks.
 

FAQ: Rolle's Theorem Problem with f(a)=f(b)=0: Finding f'(c)=f(c)/c on [a,b]

What is Rolle's Theorem?

Rolle's Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function's values at the endpoints of the interval are equal, then there exists at least one point within the interval where the derivative of the function is equal to zero.

How is Rolle's Theorem used in calculus?

Rolle's Theorem is often used in calculus to prove the existence of critical points, where the derivative of a function is equal to zero. It is also used to prove the Mean Value Theorem, which is a fundamental theorem in calculus.

What is a critical point?

A critical point is a point on a function where the derivative is equal to zero. This means that the slope of the function at that point is flat, and the function may have a maximum, minimum, or point of inflection at that point.

Can Rolle's Theorem be used to find the maximum or minimum value of a function?

No, Rolle's Theorem can only prove the existence of critical points. It cannot determine whether those points are maximum or minimum values. Additional information, such as the second derivative test, is needed to determine the nature of a critical point.

Are there any limitations to using Rolle's Theorem?

Yes, Rolle's Theorem requires that the function is continuous and differentiable on a closed interval. If these conditions are not met, then the theorem cannot be applied. Additionally, the theorem only guarantees the existence of at least one critical point, but it does not provide information about other critical points or the behavior of the function in between these points.

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