If is a real number, then what is the smallest possible value of ?

This is the equation of a parabola. Since this parabola opens upward, the vertex will be the lowest point. The x-value of the vertex is found by  -b/(2a).

a = 2    b = -8     --->     -(-8)/(2·2)  =  8/4  =  2

f(4) = 2(2)² - 8(2) + 19  =  11     <---  The minimum value of the function.