Example 4. Given the function $f(x)=x^{2}-10 x+16$.

Find the values of the function for the argument values equal to the geometric mean and arithmetic mean
of its roots, respectively.
Solution. Let's find the roots, or zeros, of the function. By equating the function to zero, we get the quadratic equation $x^{2}-10 x+16=0$, the roots of which are $x_{1}=2, x_{2}=8$. The arithmetic mean of the roots is $x^{\prime}=\frac{x_{1}+x_{2}}{2}=5$ and the geometric mean is $x^{\prime \prime}=\sqrt{x_{1} x_{2}}=4$.

Substituting the found values of the argument into the expression for $f(x)$, we get respectively

$$
f(4)=4^{2}-10 \cdot 4+16=-8, f(5)=5^{2}-10 \cdot 5+16=-9
$$