Hi Elvina.

The distance from the origin to any point (x,y) is simply sqrt(x² + y²) (you can see this by drawing a right triangle to it and using the Pythagorean Theorem).

Therefore the distance d to the point (x,y) where y = e^x can be found by calculating

d² = x² + e^2x

Note that d is always positive, so the smallest d means the smallest d². Thus you can ignore the d² (i.e. we are ignoring the square root from earlier) and just evaluate df/dx where f(x) = x² + e^2x. When you get this derivative, set it equal to zero and solve for x. This gives you the value of x at which y = e^x is closest to the origin. If you need further assistance write back.

Cheers,
Stephen La Rocque.