Math CentralQuandaries & Queries


Question from elvina, a student:

What point on the graph y = e^x is closest to the origin? Justify your answer.

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 = ex can be found by calculating

d2 = x2 + e2x

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 + e2x. When you get this derivative, set it equal to zero and solve for x. This gives you the value of x at which y = ex is closest to the origin. If you need further assistance write back.

Stephen La Rocque.

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