f-x-e-x-2-0-lt-x-lt-e-x-4-lt-x-lt-0-Find-geometric-mean-of-f-x-

Question Number 167054 by shikaridwan last updated on 05/Mar/22

f (x) = {\begin{cases} e^{- x^{2}} 0 < x < \infty \\ e^{- x^{4}} - \infty < x < 0 \end{cases}

F i n d g e o m e t r i c m e a n o f f (x)

Answered by aleks041103 last updated on 05/Mar/22

g = \lim_{a \to \infty} \lim_{Δ x \to 0} \sqrt[2 a / Δ x]{\underset{i = 0}{\prod^{\frac{2 a}{Δ x} - 1}} f (- a + i Δ x)}

\Rightarrow l n (g) = \lim_{a \to \infty} \lim_{Δ x \to \infty} \frac{Δ x}{2 a} (\underset{i = 0}{\sum^{\frac{2 a}{Δ x} - 1}} l n (f (- a + i Δ x))) =

= \underset{a \to \infty}{l i m} \frac{1}{2 a} \underset{- a}{\int^{a}} l n (f (x)) d x = l n (g)

n o w

l n (f (x)) = {\begin{cases} - x^{2}, 0 < x \\ - x^{4}, x < 0 \end{cases}

\Rightarrow g = \underset{a \to \infty}{l i m} \frac{- 1}{2 a} (\underset{- a}{\int^{0}} x^{4} d x + \int_{0}^{a} x^{2} d x) =

= \underset{a \to \infty}{l i m} \frac{- 1}{2 a} (\frac{a^{5}}{5} + \frac{a^{3}}{3}) \to - \infty

\Rightarrow l n (g) \to - \infty

\Rightarrow g = 0