1. We are given the function $K(n) = 2n^2 - 100n + 1375000$ and asked to analyze it. 2. This is a quadratic function of the form $K(n) = an^2 + bn + c$ where $a=2$, $b=-100$, and $c=1375000$. 3. To find the vertex (which gives the minimum or maximum value), use the formula for the vertex $n = -\frac{b}{2a}$. 4. Substitute $a$ and $b$: $$n = -\frac{-100}{2 \times 2} = \frac{100}{4} = 25$$ 5. Calculate $K(25)$ to find the minimum value: $$K(25) = 2(25)^2 - 100(25) + 1375000 = 2(625) - 2500 + 1375000 = 1250 - 2500 + 1375000 = -1250 + 1375000 = 1373750$$ 6. Since $a=2 > 0$, the parabola opens upward, so the vertex at $n=25$ is a minimum point. 7. Therefore, the minimum value of $K(n)$ is $1373750$ at $n=25$.