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f(x) = {(3x)/(1x)} 23x taking log on both sides log(fx) = (2 3x) log {(3x)/(1x)} log(fx) = (2 3x) log(3x) log(1x) Now differentiate f'(x) /f(x) = (2For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n;F′′(x) = ∑∞ n=1 3n(3n−1)anx3n−2 = x {1 ∑∞ n=2 an−1x 3n−} = x {1 ∑∞ n=1 anx 3n} = xf(x) 812 10 8 6 4 2 0 2105 0 05 1 15 2 25 3 x y Figure 1 Graph of the Airy function f(x) = 1 ∑∞ n=1 anx 3n Remark Solutions of the ODE y′′ = xy are called Airy functions For x large and positive, they behave like exponential functions, and for x large and
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