# Tricky Advanced Calc Problem

• Jun 5th 2009, 07:22 AM
joeyjoejoe
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 $\displaystyle \lambda$ there is a real number c $\displaystyle \in$ [a,b] such that f'(c) = $\displaystyle \lambda$f(c).

Tried to use MVT (and Rolle's) to no avail. Any suggestions?
• Jun 5th 2009, 08:27 AM
HallsofIvy
Quote:

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 $\displaystyle \lambda$ there is a real number c $\displaystyle \in$ [a,b] such that f'(c) = $\displaystyle \lambda$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 $\displaystyle F(x)= \int_a^x f(t)dt$ and apply Rolle's theorem to $\displaystyle f(x)- \lambda F(x)$.
• Jun 11th 2009, 09:54 AM
pankaj
Let $\displaystyle g(x)=e^{-\lambda x}f(x)$

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