Determine the minimum value of the function f(x)=4x^2-8x+1.

The minimum of the function is reached for the critical
value of the function. The critical value represents the root of the first derivative of
the function.

We'll determine the 1st derivative,
differentiating with respect to x:

f'(x) = 8x -
8

We'll cancel out
f'(x):

f'(x) = 0

8x - 8 =
0

8x = 8

x =
1

We'll calculate the minimum of the function for the
critical value x = 1.

f(1) = 4 - 8 +
1

f(1) = -3

The
minimum value of the function is represented by the pair:(1 ;
-3).