There are exactly three positive real numbers <img alt="$k$" height="14" src="https://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png" width="12" /> such that the function <img alt="\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]" src="https://latex.artofproblemsolving.com/0/d/b/0dbe0a5b2a8136e0b1e02478bc822fe2ba6ada41.png" style="width: 318px; height: 38px;" />   defined over the positive real numbers achieves its minimum value at exactly two positive real numbers <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />. Find the sum of these three values of <img alt="$k$" height="14" src="https://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png" width="12" />.

Question No: 451

There are exactly three positive real numbers $k$ such that the function

$f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.$

defined over the positive real numbers achieves its minimum value at exactly two positive real numbers $x$ . Find the sum of these three values of $k$ .

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Tags: Algebra

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