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