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Advanced Math / Nonlinear functions Difficulty: Hard

$f (x) = x^{2} - 48 x + 2,304$

What is the minimum value of the given function?

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Explanation

The correct answer is $1,728$ . The given function can be rewritten in the form $f (x) = a {(x - h)}^{2} + k$ , where $a$ is a positive constant and the minimum value, $k$ , of the function occurs when the value of $x$ is $h$ . By completing the square, $f (x) = x^{2} - 48 x + 2,304$ can be written as $f (x) = x^{2} - 48 x + {(\frac{48}{2})}^{2} + 2,304 - {(\frac{48}{2})}^{2}$ , or $f (x) = {(x - 24)}^{2} + 1,728$ . This equation is in the form $f (x) = a {(x - h)}^{2} + k$ , where $a equals 1$ , $h equals 24$ , and $k equals 1,728$ . Therefore, the minimum value of the given function is $1,728$ .