G(x) = x-e x/2, then g' (x) = 1-e x/2, when g'(x)=0, x=ln2, when x=ln2, the function g (x) obtains the maximum value g (LN2) = LN2.

The following is the complete process:

Analysis: ∵ point p is on the curve y = 1/2e x, and point q is on the curve y=ln(2x).

The function y = 1/2e x and the function y=ln(2x) are reciprocal functions.

Their images are symmetrical about the straight line y = x.

The distance from the point p (x, 1/2e x) to the straight line y=x is:

D=|x-y|/√2=|x- 1/2e^x|/√2

Let f (x) = (x- 1/2e x)/√ 2.

Let f' (x) = (1-1/2ex)/√ 2 = 0 = > e x = 2 = = > x=ln2

f''(x)=(- 1/2e^x)/√2==>； f "(LN2)=-√2/2 & lt； 0

∴f(x) takes the maximum value at x=ln2 (ln2- 1)√2/2.

The minimum distance from point P (x, 1/2e x) to line y=x is: (1-ln2)√2/2.

∴|pQ| The minimum value is 2 * (1-ln2) √ 2/2 = (1-ln2) √ 2.

Option b