4. Find the maximum and minimum value of the function

$$
f(x)=x^{2}-2 x-1
$$

for those real numbers $x$ that satisfy the condition $x^{4}+36 \leq 13 x^{2}$.
Solution. First, we find all $x$ for which the inequality

$$
x^{4}-13 x^{2}+36 \leq 0
$$

holds. The solutions to the biquadratic equation $x^{4}-13 x^{2}+36=0$ are: $-3,-2,2,3$. The inequality holds for $x \in[-3,-2]$ and $x \in[2,3]$.

The parabola $f(x)=x^{2}-2 x-1$ has a minimum at the point $x=1$. Therefore, it decreases on the interval $[-3,-2]$, so $14=f(-3) \geq f(x) \geq f(-2)=7$ for $x \in[-3,-2]$, and increases on the interval $[2,3]$, so $-1=f(2) \leq f(x) \leq f(3)=2$ for $x \in[2,3]$.

Accordingly, the smallest value of $f(x)$ for those $x$ for which $x^{4}+36 \leq 13 x^{2}$ holds, is -1, and the largest value is 14.

## III year