2. The graph of the quadratic function $f(x)=x^{2}+2 p x-p^{2}+7 p-2015$ intersects the coordinate axes at three points: $A, B$ and $C$. Find the value of $p$ for which the product of the lengths of the segments $O A \times O B \times O C$ is minimized.
Answer: $p=\frac{7}{2}$. Solution. For the parabola to intersect at three points, it is necessary for the equation to have two distinct roots $x_{1}$ and $x_{2}$. It is not difficult to verify that the discriminant of the given polynomial is always positive. Then the product $O A \times O B \times O C=\left|x_{1}\right| \cdot\left|x_{2}\right| \cdot|f(0)|$. Applying Vieta's theorem, we get that this product is equal to the square of the constant term. Therefore, it is minimized when $\left(-p^{2}+7 p-\right.$ $2015)^{2}$ is minimized, which occurs at $p=\frac{7}{2}$.