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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
The height of such a rectangle will then by
The area of such a rectangle is then a function of
Differentiating with respect to
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
This means the largest area of this rectangle is