Let the coordinates of point P be (x, e^x). The area of the triangle is base times height over 2. Let us call AB the base, and the height will be the perpendicular extending from point P onto the segment AB. (Hope you're following so far.)
First we measure the distance AB using the distance formula:
For the height, we need to first find the equation of the line perpendicular to AB that passes through P. The slope of this line is the negative reciprocal of the slope of AB, which is
Find the equation of the line containing PH, where PH is the height and H is on AB.
Use this equation to find the coordinates of H. (Intersect the two lines, which algebraically means: set the equations equal.)
Finally, apply the distance formula to find PH and set up the formula for the area of the triangle. Remember to only consider critical number between 0 and 1, and justify that the area is an optimum of the type you want (i.e. max and not min)