B 1 = (1-x) * e x, so m is (0, (1-x) * e x).

The vertical slope k2 =- 1/e x, let a straight line pn: y = k2x+B2 and replace it with p.

B2 = e x+x/e x, so n is (0, e x+x/e x).

So t =1/2 * [(1-x) * ex+ex+x/ex], take the derivative, and make t'=0, which simplifies it.

(1-x) (1/e x+e x) = 0, because ex >; 0, 1/e^x>； 0, so only 1-X=0.

Get X= 1, and substitute t to get the minimum value 1/2(e+ 1/e).