If the critical value is not in {{MATH_205}} it is larger than {{MATH_206}}. In this case the minimum must occur at one of the endpoints. We can compute {{ENV_207}} but it is difficult to determine which of these is smaller by direct comparison. If, as is likely in practice, we know the values of {{MATH_208}}, {{MATH_209}}, {{MATH_210}}, and {{MATH_211}}, then it is easy to determine this. With a little cleverness, however, we can determine the minimum in general. We have seen that {{MATH_212}} is always positive, so the derivative {{MATH_213}} is always increasing. We know that at {{MATH_214}} the derivative is zero, so for values of {{MATH_215}} less than that critical value, the derivative is negative. This means that {{MATH_216}}, so the minimum occurs when {{MATH_217}}.