FILM 1900 Lecture : Producing and Pre-Production

Show, for a positive integer n, that F^n = W_1⊕W_2, where W_1 and W_2 are the following subspaces of F^n:

W_1 ={(a1,...,an) ∈ Fn | an = 0}

W_2 ={(a1,...,an) ∈ Fn | a1 = a2 = ··· = an−1 = 0}.

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).