Question

Let f(x) ∈ C[0, 1] ∩ C 1 [0, 1], and f 0 (x) > 0, f'(0) = 0. Prove that there exist λ ∈ (0, 1) and µ ∈ (0, 1) such that λ + µ = 1 and f ' (λ) / f(λ) = f '(µ) / f(µ). Recall that Ck (I) is the set of all continuous functions which have continuous derivatives up to order k in the interval I.

Answer 1

The problem is to prove the existence of λ and μ such that λ + μ = 1 and f'(λ) / f(λ) = f'(μ) / f(μ) under given conditions. The Mean Value Theorem or Rolle's Theorem may be used to establish this relationship due to the properties of f given in C[0, 1] ∩ C1 [0, 1].

Step-by-step explanation:

The student's question revolves around the existence of two numbers λ and μ within the interval (0,1) such that λ + μ = 1 and the ratio of the derivative of f to f itself at these points is equal, given certain conditions on f. The function f is continuous on the interval [0,1] and also has a continuous first derivative on the same interval, which is represented mathematically as f(x) ∈ C[0, 1] ∩ C1 [0, 1]. Note that f'(0) = 0 suggests that x = 0 might be an extremum, and f0(x) > 0 ensures the function is positive on (0,1). To prove that such λ and μ exist, we would appeal to the Mean Value Theorem or Rolle's Theorem, as these provide a relationship between function values and derivatives.