Let ABCD be the rectangle inscribed in a semicircle of radius 1 unit such that the vertices A and B lie on the diameter.

Let AB = DC = x and BC = AD = y.

Let O be the centre of the semicircle.

Join OC and OD. Then OC = OD = radius = 1.

Also, AD = BC and m∠A = m∠B = 90°.

∴ OA = OB

∴ OB = 1/2 AB = x/2

In right angled triangle OBC,

OB^{2} + BC^{2} = OC^{2}

For maximum value of f(x), f'(x) = 0

∴ by the second derivative test, f is maximum when x = **√**2

Hence, the area of the rectangle is maximum (**i.e.** rectangle has the largest size) when its length is √2 units and breadth is 1/√2 unit.