Math Help - Tricky Advanced Calc Problem

1. Tricky Advanced Calc Problem

Suppose that f is continuous and differentiable on [a,b], f(a) = f(b) = 0. Show that for every real number $\lambda$ there is a real number c $\in$ [a,b] such that f'(c) = $\lambda$f(c).

Tried to use MVT (and Rolle's) to no avail. Any suggestions?

2. Originally Posted by joeyjoejoe
Suppose that f is continuous and differentiable on [a,b], f(a) = f(b) = 0. Show that for every real number $\lambda$ there is a real number c $\in$ [a,b] such that f'(c) = $\lambda$f(c).

Tried to use MVT (and Rolle's) to no avail. Any suggestions?
Since F is continuous on [a, b] it is integrable there. Let $F(x)= \int_a^x f(t)dt$ and apply Rolle's theorem to $f(x)- \lambda F(x)$.

3. Let $g(x)=e^{-\lambda x}f(x)$

g(a)=g(b)=0.Now use Rolle's Theorem