2-3. Find the minimum value of the function

$$
f(x)=(x-1)^{2}+(x-3)^{2}+\ldots+(x-101)^{2}
$$

If the result is a non-integer, round it to the nearest integer and write it in the answer.
Solution. $f(x)=51 x^{2}-2(1+3+\ldots+101) x+\left(1^{2}+3^{2}+\ldots+101^{2}\right)=51 x^{2}-2 \cdot 51^{2} x+\left(1^{2}+3^{2}+\ldots+101^{2}\right)$.

The minimum of the function $f$ is achieved at the point $x_{0}=51$. Since $1^{2}+3^{2}+\ldots+(2 n-1)^{2}=\frac{n\left(4 n^{2}-1\right)}{3}$, then $f(51)=51^{3}-2 \cdot 51^{3}+51^{3}+\frac{51^{3}}{3}-\frac{51}{3}=\frac{51^{3}}{3}-\frac{51}{3}=44200$.

Answer: 44200.