Minimize the distance from a curve:

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Find the coordinates of a point on the curve which is closest to the point (5,0).

We can minimize the square of the distance, just as well as the distance itself. We will have an easier equation to work with as a result:

Since the point must be on the given curve, the y coordinate can be replaced by its equivalent in terms of x. Hence we can express the square of the distance entirely in terms of x:

In order to minimize this, we differentiate and set the result to 0. First, the derivative is

If we set this to 0 to find the coordinates of the closest point , we get

, or

We can find the corresponding y coordinate from the equation of the curve:

Hence (4.5, 2.12) are the coordinates of the desired closest point on the original curve to the point (5,0). We can find the actual distance by substituting in the equation for the square of the distance and taking the square root:

, or

Test Problem: Find the coordinates of the point on closest to (6, 0).