Choice B is correct. It's given that $f (x) = (x - 44) (x - 46)$ , which can be rewritten as $f (x) = x^{2} - 90 x + 2,024$ . Since the coefficient of the $x^{2}$ -term is positive, the graph of $y = f (x)$ in the xy-plane opens upward and reaches its minimum value at its vertex. For an equation in the form $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, the x-coordinate of the vertex is $- \frac{b}{2 a}$ . For the equation $f (x) = x^{2} - 90 x + 2,024$ , $a equals 1$ , $b = - 90$ , and $c equals 2,024$ . It follows that the x-coordinate of the vertex is $- \frac{(- 90)}{2 (1)}$ , or $45$ . Therefore, $f (x)$ reaches its minimum when the value of $x$ is $45$ .

Choice A is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y = f (x)$ in the xy-plane.

Choice C is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y = f (x)$ in the xy-plane.

Choice D is incorrect. This is the y-coordinate of the vertex of the graph of $y = f (x)$ in the xy-plane.