My assignment says the perimeter of a rectangle is 30 cm.
find the whole-number dimensions of the rectangle with:
a) the greatest area
b) the least area
can someone explain?

Let side be "a" cm. So another side is (30-2a)/2=15-a
S=ab=a(15-a)=15a-a^2 - parabola
maximum when  a =  -15/-2=7.5 cm 
When a=7.5 all sides are 7.5 cm 
So it's a square. S=a*a=7.5*7.5=56.25 cm^2 - max
There is no global minimun for S.(look at the graphic)

A. is the answer in this question