Find the largest possible area of a rectangle with base on x-axis and vertices on curve y = 4 - x^2

The region bounded by the parabola and the x-axis is symmetric, so any inscribed rectangle whose base lies in the x-axis will have its base extend symmetrically around the origin, i.e. if the base has length , with , then its base is the line segment connecting the points and .

The height of such a rectangle will then by .

The area of such a rectangle is then a function of :



Differentiating with respect to gives



which has critical points at



We omit the negative root. Checking the sign of the second derivative at the positive critical point (it's negative) confirms that is the site of a local maximum.

This means the largest area of this rectangle is

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