Maximum area of the rectangle is [tex]9cm^{2}[/tex]
Explanation:
Considering the dimensions to be in cm
[tex]f(x) = -(x-3)^{2} +9\\f(x) = -(x^{2} +9 - 6x)+9\\f(x) = -x^{2} +6x\\f'(x) = -2x+6\\-2x+6 = 0\\2x=6\\x=3cm\\\\[/tex]
Putting the value of x = 3
[tex]Perimeter = 2(x+b)\\12 = 2(3+b)\\6 = 3+b\\b= 3cm[/tex]
[tex]Area of rectangle = x X b\\ = 3 X 3\\ = 9cm^{2}[/tex]
Therefore, maximum area of the rectangle is [tex]9cm^{2}[/tex]
