Question

Let b > 0 and f : [0, b] → R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n ∈ N, there exists some c ∈ (0, b) such that cf ′(c) = −nf(c).

Hint: Apply Rolle’s theorem to the function g(x) = xⁿ f(x).

5.2.10. Let b > 0 and f : [0, b] → R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n N, there exists some c (0, b) such that cf,(c) -nf(c). Hint: Apply Rolle's theorem to the function g(x)f(x)