Then {{MATH_122}} where {{MATH_123}} Therefore, we can express the area {{MATH_124}} as a function of {{MATH_125}} {{MATH_126}} We want the maximum value of {{MATH_127}} when {{MATH_128}} is in {{MATH_129}}. You might object to allowing {{MATH_130}} or {{MATH_131}}, since then the ``rectangle'' has either no width or no height, so is not ``really'' a rectangle. But the problem is somewhat easier if we simply allow such rectangles, which have zero area as we may then apply the Extreme Value Theorem and see that we indeed have a maximum and minimum value. {{CMD_132}} {{CMD_133}}At this point, you should graph the function if you can.{{CMD_134}} {{CMD_135}} Setting {{MATH_136}} we find {{MATH_137}} as the only critical point. Testing this and the two endpoints (as the maximum could also be there), we have {{MATH_138}} and {{MATH_139}}. Hence, the maximum area occurs when the rectangle has dimensions {{MATH_140}}. {{CMD_141}} {{CMD_142}}