Prove the inequality (15.30). Hint: The quantity (15.29) is invariant with respect to scale. By taking 2

Prove the inequality (15.30). Hint: The quantity (15.29) is invariant with respect to scale. By taking σ2 = 1, the problem reduces to choosing f to minimize  f 2 subject to f being a mean 0 density with variance 1. Using the method of undetermined multipliers, it is sufficient to minimize

[ f 2

(x) + 2b(x 2 − a2

) f (x)]dx , where a and b are chosen so that f is a mean 0 density with variance 1.