EXERCISE 2

A rectangle has perimeter p and area a:

a) If the perimeter is 8 find the dimensions which give the maximum possible area.

• Let x be the base of the rectangle. Look carefully at the unknown factors in the window and write a function in the form y=f(x) in order to find the maximum.

• Find f'(x) and solve the equation: f'(x)=0

• Find f''(x) and its sign at these values.

Look carefully at the curves of f´ and f'' in the window.

• Where does f'(x) cut the X-axis? What is the behaviour of f''(x) at these points?

Change the value of x and the curve of y=f(x) will appear so that you can check your results.

 b) If the area is 4 find the dimensions which give the minimum possible perimeter.

Look carefully at the unknown factors in the window. Which function do we now need in order to find its minimum? Write it down.

• Where does f'(x) cut the X-axis? Are both solutions valid in this case? What is the sign of f''(x) at these points?

When you change the value of x you can see the graph of f(x) being drawn.

• As above, find f'(x) and solve the equation: f'(x)=0

• Find f''(x) and its sign at these values