MATH 295 PROBLEMS 13

IGOR KRIZ

1. Prove that it is impossible to write x = f(x)g(x) where f and g are defined and differentiable (have derivative) in an interval (−ϵ, ϵ), ϵ > 0, and satisfy f(0) = g(0) = 0. [Hint: differentiate.]

2. Prove that of all rectangles with the same perimeter, the square has the greatest area. [Hint: express the problem in terms of finding the maximum of a function, then use derivatives.]

3. Let f : ℝ → ℝ be a function. Suppose that f′(x) > g′(x) for all x, and f(a) = g(a). Show that f(x) > g(x) for all x > a and f(x) < g(x) for all x < a.

4. Let f : ℝ → ℝ be a function. Prove that if

lim_{x→+∞} f(x) and lim_{x→+∞} f′(x)

both exist, then

lim_{x→+∞} f′(x) = 0.