B3. Solve the given problems: a) Calculate the intersections and the size of the angle between the curves $y=-x^{-2}$ and $y=-\frac{x^{2}}{2}+\frac{1}{2}$. b) The function $f(x)=\frac{x}{\ln x}$ is given. Determine the domain of the function $f$ and the intervals where the function $f$ is increasing and decreasing.

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 23, 2022  Solutions to Problems for 1st Year

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ |
| :--- | :---: | :---: |
| $\mathrm{D}$ | $\mathrm{D}$ | $\mathrm{B}$ |
B3. a) We equate the curves $-x^{-2}=-\frac{x^{2}}{2}+\frac{1}{2}$, then multiply by the common denominator and rearrange to the form $x^{4}-x^{2}-2=0$. We write it in the form of a product $\left(x^{2}-2\right)\left(x^{2}+1\right)=0$, then factor the first parenthesis to $(x-\sqrt{2})(x+\sqrt{2})\left(x^{2}+1\right)=0$. We get the solutions $x_{1}=\sqrt{2}$ and $x_{2}=-\sqrt{2}$. We calculate the corresponding ordinates and write the intersection points $P_{1}\left(\sqrt{2},-\frac{1}{2}\right)$ and $P_{2}\left(-\sqrt{2},-\frac{1}{2}\right)$. We differentiate the first curve $y^{\prime}=2 x^{-3}$ and write its slope $k_{1}=\frac{\sqrt{2}}{2}$. We differentiate the second curve $y^{\prime}=-x$ and write its slope $k_{2}=-\sqrt{2}$. We substitute the data into the formula for calculating the angle between the curves and get tan $\left.\alpha=\left|\frac{k_{2}-k_{1}}{1+k_{1} \cdot k_{2}}\right||| \frac{-\sqrt{2-\frac{\sqrt{2}}{2}}}{1+\left(\frac{\sqrt{2}}{2}\right)(-\sqrt{2})} \right\rvert\,=\infty$. The angle between the curves is $\alpha=90^{\circ}$.

b) The function is not defined at the pole at $x=1$. We write the domain of the function $D_{f}=$ $(0,1) \cup(1, \infty)$. We differentiate the function and get the derivative $f^{\prime}(x)=\frac{\ln x-1}{(\ln x)^{2}}$. We calculate the zero of the derivative $\ln x=1$ and get the solution $x=e$. The pole of the function is at $x=1$. We determine the interval of increase $(e, \infty)$ and the interval of decrease $(0,1) \cup(1, e)$.
Solving the equation $-x^{-2}=-\frac{x^{2}}{2}+\frac{1}{2}$ ..... 1 point
Calculating $k_{1}=\frac{\sqrt{2}}{2}$ and $k_{2}=-\sqrt{2}$ ..... 1 point
Writing the intersection points $P_{1}\left(\sqrt{2},-\frac{1}{2}\right)$ and $P_{2}\left(-\sqrt{2},-\frac{1}{2}\right)$ ..... 1 point
Calculating the angle between the curves $\alpha=90^{\circ}$ ..... 1 point
Writing the domain of the function $D_{f}=(0,1) \cup(1, \infty)$ ..... 1 point
Writing the derivative of the function $f^{\prime}(x)=\frac{\ln x-1}{(\ln x)^{2}}$ ..... 1 point
Writing the interval of increase $(e, \infty)$ ..... 1 point
Writing the intervals of decrease $(0,1) \cup(1, e)$ ..... $1^{*}$ point