Find the smallest function value y = 4×2 – 4x + 3.

We transform this expression by selecting a full square:

y = 4 * x2 – 4 * x + 3 = (2 * x) ^ 2 – 2 * (2 * x) * 1 + (1) ^ 2 – (1) ^ 2 + 3 = [2 * x – 1 ] ^ 2 – 1 + 3 = (2 * x – 1) ^ 2 + 2. (1)

When selecting a complete square, the square of the second number was added – (1) ^ 2, and so that the expression did not change, the same (1) ^ 2 was subtracted.

This means that the desired function y took the form: y = (…) ^ 2 + 2. The expression in parentheses in the square is equal to or> 0, and the minimum value in parentheses is 0. Hence, expression (1) takes the minimum value equal to 2 …