# real analysis

• May 7th 2010, 04:21 AM
abk6690
real analysis
1a) suppose that f is continuous on [a b] and differentiable on (a b) with f(a) = f(b) = 0. Show that for every k in R, there exists a c in (a b) with f '(c) = kf(c). (Hintconsider the function g(x) = exp^(-kx)f(x) for x in [a b]

b) sho that f(x) = sinx is unformly continuos on R. (Hint: if |x-y|< delta, use the Mean Value Theorem on [x y].)
• May 7th 2010, 06:30 AM
Krizalid
your problems are already solved, wonder why you didn't attach any attempt of solution.

given $g(x)=e^{-kx}f(x),$ since $g(a)=g(b)=0$ then exists $c\in(a,b)$ so that $g'(c)=-ke^{-kc}f(c)+e^{-kc}f'(c)=0\implies f'(c)=kf(c).$

the second one has been proved many times here, i suggest to you to use the seach engine.