What is the x-coordinate of the point on the curve y=e^x that is closest to the orgin?

Answer 1

Let the point of contact be P(x,y)
and the distance from the origin be D
D^2 = x^2 + y^2
= x^2 + e^(2x)
2D dD/dx = 2x + 2e^(2x)
= 0 for a min of D
so
x = -e^(2x) , a nasty equation to solve

I tried some trial and error, with 3 columns
x --- Right side
-.5 -.36
-.4 -.449
-.39 -.458
-.41 -.440
-.42 .4317
-.43 .4232
-.425 -.427

so x = appr. -.425