The Wronskian of a pair of differentiable functions f(x). g(x) is the scalar function

(a) Prove that if /. g are linearly dependent, then W[f(x). g(x)] = 0.
(b) Prove that if W[f(x). g(x) ‰  0, then f, g are linearly independent.
(c) Let f(x) = x3, g(x) = |x|3. Prove that f. g ˆˆ C2 are twice continuously differentiable and linearly independent, but W[f(x), g(x)] ‰¡ 0. Thus, the Wronskian is not a fool-proof test for linear independence.

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f(x) g'(x) (2.16)