# Problem 1. (2 points)

A cubic polynomial has three roots. The greatest value of the polynomial on the interval $[4 ; 9]$ is achieved at $x=5$, and the smallest value is achieved at $x=7$. Find the sum of the roots of the polynomial.

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# Answer: 18

## Solution:

Since in all options the minimum and maximum on the interval are not reached at its ends, they are reached at the roots of the polynomial's derivative. Let the polynomial be of the form $a x^{3}+b x^{2}+c x+d$, and its derivative, respectively, $3 a x^{2}+2 b x+c$. In this case, the sum of the roots of the polynomial is $\frac{-b}{a}$, and the sum of the roots of the derivative is $\frac{-2 b}{3 a}$, which is $\frac{2}{3}$ of the sum of the roots of the polynomial.

Therefore, the sum of the roots of the polynomial is $\frac{3}{2}\left(x_{1}+x_{2}\right)$.

| $x_{0}$ | 4 |
| :---: | :---: |
| $x_{1}$ | 5 |
| $x_{2}$ | 7 |
| $x_{3}$ | 9 |
| Answer | 18 |