Chapter 6, Problem 69RE

(a)

To determine

To sketch: The function $f\left(x\right)=\frac{1}{x\sigma \sqrt{2\pi }}{e}^{\frac{-{\ln }^{2}x}{2{\sigma }^2}},x\text{ >0}$ for $\sigma =\frac{1}{2},1,2$ and to find whether the limit $\underset{x\to 0^{+}}{\lim }f\left(x\right)$ exists or not.

(b)

To determine

To evaluate: The value of $\underset{x\to 0^{+}}{\lim }f\left(x\right)$.

(c)

To determine

To show: f has a single local maximum at $x^{*}=e^{-{\sigma }^2}$.

(d)

To determine

To evaluate: $f\left(x^{*}\right)$ and express the result as a function of $\sigma$.

(e)

To determine

To find: The value of $\sigma >0$ such that $f\left(x^{*}\right)$ is minimum.