Differentiation - Problems Involving Maximisation and Minimisation

The diagram shows the planning diagram of a garden that consists of two identical equilateral triangles and a rectangle, where the side of the equilateral triangles are \(x\) m and the width of the rectangle is \(l\) m. The owner uses a total of \(160\) m of fencing to enclose the garden.

1. Show that the area, \(A\) m\(^2\), of the garden, is given by \(A = \frac{160x - (4 - \sqrt{3})x^2}{2}\).
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
3. Explain why this value of \(x\) maximises the value of \(A\).