In a school, a auditorium was used for its cultural activities. The shape of the floor of the auditorium is rectangular with dimensions x and y (x > y), has fixed parameter p.

Based on the above information answer the following questions.

The value of x, for which area of floor of auditorium is maximum is :

$\frac{p}{4}$

$A'(x) = \frac{p-4x}{2} = 0$

⇒ $x = \frac{p}{4}$

so $A''(x) = \frac{-4}{2} < 0$

so $A(\frac{p}{4}) = \frac{p × \frac{p}{4} - 2 × \frac{p^2}{16}}{2}$

= $\frac{p^2}{16}$

at  $x =\frac{p}{4}$ (max area)