Problem 10.2. Find the maximum of the function

$$
f(x)=\frac{\lg x \cdot \lg x^{2}+\lg x^{3}+3}{\lg ^{2} x+\lg x^{2}+2}
$$

and the values of $x$, when it is attained.

Ivailo Kortezov
10.2. The domain of $f(x)$ is $x>0$. Setting $y=\lg x$ gives

$$
F(y)=\frac{2 y^{2}+3 y+3}{y^{2}+2 y+2}
$$

Since the denominator is positive the function $F(y)$ is defined for all real $y$.

Let $M$ be the desired value of $f(x)$ (if it exists). Then for any real $y$ we have

$$
\begin{gathered}
\frac{2 y^{2}+3 y+3}{y^{2}+2 y+2} \leq M \\
2 y^{2}+3 y+3 \leq M y^{2}+2 M y+2 M \\
(2-M) y^{2}+(3-2 M) y+(3-2 M) \leq 0
\end{gathered}
$$

Therefore $2-M<0$ and $D=(3-2 M)(3-2 M-8+4 M)=(3-2 M)(2 M-5) \leq$ 0 . Hence $M \geq 2.5$.

Note that for $M=2.5$ the above inequality becomes $-0,5 y^{2}-2 y-2 \leq 0$, i.e. $y^{2}+4 y+4=(y+2)^{2} \geq 0$ and the equality is attained only if $y=-2$, i.e. for $x=0,01$.

Therefore the maximum of the function equals 2,5 and it is attained for $x=0,01$ only.