frankfiers
New member
- Joined
- Oct 29, 2006
- Messages
- 1
The Question:
Suppose that a twice differentiable function f satisfies f"(x) + f'(x)g(x) - f(x) = 0 for some function g. Prove that if f is zero at two points, then f is zero on the interval between them.
It's a tough question. I thought that if I could show f'(x) is constant and equals zero throughout the interval then f(x) will have to be zero on the interval. f'(x) is constant if f"(x) = 0 throughout the interval. So what I think I have to show is f(x) = f'(x) = f"(x) = 0.
Using Rolle's Theorem, I can find a point at "c" where f'(x) = 0. And at that point: f"(c) - f(c) = 0. But here is where I am lost. I am not certain if my intial lead is any good. I think that if I can show f(c) = 0, then I can show by using Rolle's Theorem again that there is another point between "c" and the endpoint of the interval that has value f(x) = 0, and so on and so on, proving that indeed the f is zero on the interval.
If anyone can solve the problem, please share the solution.
Much thanks,
Frank Fiers.
Suppose that a twice differentiable function f satisfies f"(x) + f'(x)g(x) - f(x) = 0 for some function g. Prove that if f is zero at two points, then f is zero on the interval between them.
It's a tough question. I thought that if I could show f'(x) is constant and equals zero throughout the interval then f(x) will have to be zero on the interval. f'(x) is constant if f"(x) = 0 throughout the interval. So what I think I have to show is f(x) = f'(x) = f"(x) = 0.
Using Rolle's Theorem, I can find a point at "c" where f'(x) = 0. And at that point: f"(c) - f(c) = 0. But here is where I am lost. I am not certain if my intial lead is any good. I think that if I can show f(c) = 0, then I can show by using Rolle's Theorem again that there is another point between "c" and the endpoint of the interval that has value f(x) = 0, and so on and so on, proving that indeed the f is zero on the interval.
If anyone can solve the problem, please share the solution.
Much thanks,
Frank Fiers.
