B3. Dani are real functions $f(x)=x+1$ and $g(x)=x^{2}+3$.

a) The function $h$ is given by the formula $h(x)=\frac{f(x)+g(x)}{g(x)}$. Calculate the stationary points of the function $h$. Write the equation of the horizontal asymptote of the graph of the function $h$ and calculate the intersection of the graph with the horizontal asymptote.

b) Calculate for which $a \in \mathbb{R}$ the function $j(x)=g(x)+a f(x)-a$ will have at least one real root.

## 19th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 13, 2019

## Solutions to Problems and Scoring

(April 5, 2019, 10:30)

A competitor who arrives at the solution by any correct method (even if the scoring guide does not anticipate it) receives all possible points.

A correct method is considered any procedure that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points anticipated. If, however, the solution is guessed (it is not possible to arrive at it without calculation), even if it is correct, it is scored with 0 points.

A competitor who has only partially solved the problem, from a generally correct solving procedure but without a clear path to the final solution of the problem, cannot receive more than half of the possible points.

The asterisk '*' next to the points means that the competitor can receive the point or points for a correct procedure, even if the calculation is incorrect.

## First Year

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-19.jpg?height=137&width=220&top_left_y=340&top_left_x=929)
B3.

a) We write the function rule $h(x)=\frac{x^{2}+x+4}{x^{2}+3}$. We calculate the derivative of the function $h^{\prime}(x)=\frac{-x^{2}-2 x+3}{\left(x^{2}+3\right)^{2}}$. The zeros of the derivative of the function $h$ are the solutions of the equation $-x^{2}-2 x+3=0$. The solutions of the equation are $x_{1}=1, x_{2}=$ -3, which are the stationary points of the function $h$. The equation of the horizontal asymptote of the graph of the function $h$ is $y=$ 1. The abscissa of the intersection of the graph of the function $h$ with the horizontal asymptote is the zero of the remainder when the numerator is divided by the denominator of the rational function $h$. The zero of the remainder $r(x)=x+1$ is $x=-1$. The intersection is the point $P(-1,1)$.

b) We write the function rule $j(x)=x^{2}+a x+3$. The function $j$ will have at least one real zero if $D \geq 0$, which means $a^{2}-12 \geq 0$. The solutions of the inequality are $-2 \sqrt{3} \geq a$ or $a \geq 2 \sqrt{3}$.
Writing $h(x)=\frac{x^{2}+x+4}{x^{2}+3}$ ..... 1 point
Calculating $h^{\prime}(x)=\frac{-x^{2}-2 x+3}{\left(x^{2}+3\right)^{2}}$ ..... 1 point
Writing the equation $-x^{2}-2 x+3=0$ ..... 1 point
Writing the stationary points $x_{1}=1, x_{2}=-3$ ..... 1 point
Writing the equation of the horizontal asymptote $y=1$ ..... 1 point
Writing the intersection $P(-1,1)$ ..... 1 point
Writing the inequality $a^{2}-12 \geq 0$ ..... 1 point
Result $-2 \sqrt{3} \geq a$ or $a \geq 2 \sqrt{3}$ ..... 1 point