8. What is the area of the largest rectangle that can be inscribed in the region bounded by y = 3 – x² and the x-axis?

Consider the following graph of the region described above with an inscribed rectangle of length L = 2x and width W = y = 3 – x²:

Since the area A, of any rectangle, is given by the product of its length and width,

So that the maximum area is given by, evaluating A(x) at the critical points and choosing the maximum value. The critical points are found by solving the equation, A’(x)=0, as follows:

Hence, the largest rectangle that can be inscribed in the above region is,