
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 xy< delta, use the Mean Value Theorem on [x y].)

your problems are already solved, wonder why you didn't attach any attempt of solution.
given $\displaystyle g(x)=e^{kx}f(x),$ since $\displaystyle g(a)=g(b)=0$ then exists $\displaystyle c\in(a,b)$ so that $\displaystyle g'(c)=ke^{kc}f(c)+e^{kc}f'(c)=0\implies f'(c)=kf(c).$
the second one has been proved many times here, i suggest to you to use the seach engine.