Area can be expressed as a quadratic function of the base (x) under the
condition: x + 2h = 10, where h is the height
of the rectangle. By calculating the coordinates of the vertex we get the
result. It can be proved by writing the quadratic equation into GeoGebra's
Input Bar to get the graph. If our equation is correct, the points from
section 3 will lie on it. By using the command Extreme the coordinates of
vertex will appear in the algebraically window [or following similar procedure
with TI-Nspire™]
Here the screencast with the approach shown in GeoGebra is presented.