3. Solve the equation

$2 x+1+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+1) \cdot \sqrt{x^{2}+2 x+2}=0$.
Answer: $-\frac{1}{2}$.

Solution. Let $f(x)=x+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}$. The original equation can be rewritten as $f(x)+f(x+1)=0$. Note that the function $f(x)$ is odd. It increases on the positive half-axis (as the sum of increasing functions). Due to its oddness, it increases on the entire real line. Further, we have

$$
f(x)=-f(x+1) ; \quad f(x)=f(-x-1)
$$

Since an increasing function takes each of its values exactly once, it follows that $x=-x-1$, from which $x=-\frac{1}{2}$.

Remark. The monotonicity of the left side of the original equation could also have been established using the derivative.

Evaluation. 12 points for a correct solution. If the answer is guessed but not proven that there are no other solutions, 3 points.