We know a real number satisfies where is positive. Let be an approximation of such that the relative error,
is minimized. Denote the maximum of this relative error by as varies over the interval.
- Prove that has its maximum at one of the end points.
- Prove that is minimal when
This value of is called the harmonic mean of and .
- Proof. We rewrite the error term as a piecewise function (so we can consider the derivative of each piece),
So, taking the derivative with respect to we then have,
But we know from the problem statement that and since both and are in the interval we have . Therefore, the derivative is not zero anywhere on the interval (since and since is not zero). Hence, cannot have a maximum on the interior , so the maximum must be at one of the endpoints, or
- Proof. Now, we want to find the value of at which is smallest (i.e., when the maximum error is smallest). From part (a), we know that for any fixed the maximum value of the error must occur at an end point, or . Since is the function which returns the maximum error, we have
Since this is the maximum error, we then have
Furthermore, since we know that and . Now,
Hence, we can give a formula for ,
Taking the derivative of with respect to we then have
Since and are both positive, this means is negative if and is positive if ; hence, is decreasing for and is increasing for . Therefore, has a minimum at