The function given is : f(x) = 4x^2 + 4x + 1
4x^2 + 4x + 1
= (2x)^2 + 2*2x*1 + 1^2
= (2x + 1)^2
The function f(x) = 4x^2 + 4x + 1 = (2x + 1)^2. The square of real numbers is always positive and the minimum value it can take on is 0^2 =0.
The function f(x) cannot take on values lower than 0, as the square of negative as well as positive real numbers is positive.