
Problem 11.1 Let $f(x)=\frac{x^{2}+4 x+3}{x^{2}+7 x+14}$.
a) Find the greatest value of $f(x)$;

b) Find the greatest value of the function $\left(\frac{x^{2}-5 x+10}{x^{2}+5 x+20}\right)^{f(x)}$.


Solution: a) We shall prove that the greatest value of $f(x)$ equals 2. Since $x^{2}+7 x+14>0, \forall x$, we obtain $f(x) \leq 2 \Longrightarrow(x+5)^{2} \geq 0$, and an equality occurs only when $x=-5$.

b) Let $g(x)=\frac{x^{2}-5 x+10}{x^{2}+5 x+20}$. Since $x^{2}+5 x+20>0, \forall x$, we obtain $g(x) \leq 3 \Longrightarrow(x+5)^{2} \geq 0$, and equality occurs only when $x=-5$. Since $x^{2}-5 x+10>0, \forall x$, we have that the function $h(x)=g(x)^{f(x)}$ is correctly defined. Further, when $f(x)>0$, i.e. $x \in[-3,-1]$, then $h(x) \leq 3^{2}=9$. From the other hand $g(x) \geq 1 \Longrightarrow x \leq-1$ and so $h(x) \leq 1$ when $f(x) \leq 0$. Therefore the greatest value of $h(x)$ equals 9 and $h(x)=9$ when $x=-5$.
