52 (1180). Find the maximum value of the function $y=\frac{x}{x^{2}+4}$.
Solution. Method I. The domain of the given function is the set of all real numbers. Since for $x0$ (the function $y(x)$ is decreasing) when $x^{2}-4>0$.

Thus, the function $y(x)$ increases on the interval $[-2 ; 2]$, and decreases on the intervals $(-\infty ;-2]$ and $[2 ;+\infty)$.

The maximum value of the function is attained at $x=2$.

$$
y(2)=\frac{2}{4+4}=\frac{1}{4}
$$

Method II. We will find the maximum value of the function $y$ experimentally. Let's find several values of the given function for $x \geqslant 0$ (since the maximum value of the function $y$ is achieved for $x \geqslant 0$):

$$
y(0)=0, y(1)=\frac{1}{5}, y(2)=\frac{1}{4}, y(3)=\frac{3}{10}, y(4)=\frac{1}{5}
$$

Among the found values of $y$, the largest is $\frac{1}{4}$. We will prove that $\frac{1}{4}$ is the maximum value of the function over its entire domain, i.e., we will prove that for all $x$ the inequality $\frac{x}{x^{2}+4} \leqslant \frac{1}{4}$ holds.

We have $4 x \leqslant x^{2}+4$, from which $x^{2}-4 x+4 \geqslant 0$. We obtained the obvious inequality $(x-2)^{2} \geqslant 0$. The statement is proved.

Answer: $\frac{1}{4}$.

Remark. The considered method is not universal and, undoubtedly, will not always lead to success. However, in some cases it can be useful and, above all, for developing heuristic thinking in students.