Question 15
There are exactly three positive real numbers $k$ such that the function $$
\begin{aligned} f(x) &= \frac{(x - 18)(x - 72)}{x} \\ &\quad {}\cdot (x - 98)(x - k) \end{aligned}
$$ 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 $k.$
\begin{aligned} f(x) &= \frac{(x - 18)(x - 72)}{x} \\ &\quad {}\cdot (x - 98)(x - k) \end{aligned}
$$ 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 $k.$