If f(x)=( 1^ )x+ _e(123)x _e(1234)-( 1^ ),x0, then the least value of f(f(x))+f(f((4)/(x)…

Question

If $f(x)=\frac{(	an {1}^{\circ })x+{\log }_{e}(123)}{x{\log }_{e}(1234)-(	an {1}^{\circ })},x>0$, then the least value of $f(f(x))+f(f(\frac{4}{x}))$ is :

Accepted Answer

Given that $f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _e(123)}{x \log _e(1234)-\left(\tan 1^{\circ}\right)}$

Let us consider a similar function of $(x)$,

$\therefore f(x)=\frac{A x+B}{C x-A}$

$\text { Now, } $

$$
\begin{aligned}
&f(f(x)) =\frac{A\left(\frac{A x+B}{C x-A}\right)+B}{C\left(\frac{A x+B}{C x-A}\right)-A} \\
& =\frac{A^2 x+A B+B C x-A B}{A C x+B C-A C x+A^2} \\
& =\frac{x\left(A^2+B C\right)}{\left(B C+A^2\right)}=x
\end{aligned}
$$

$$
\begin{aligned}
& \therefore \quad f(f(x))=x \\
& \text { Similarly, } f\left(f\left(\frac{4}{x}\right)\right)=\frac{4}{x} \\
& \text { Apply, AM } \geq \text { GM } \\
& \left(x+\frac{4}{x}\right) \geq 2 \sqrt{x \cdot \frac{4}{x}}=4(x>0) \\
& \Rightarrow f(f(x))+f\left(f\left(\frac{4}{x}\right)\right) \geq 4
\end{aligned}
$$

Hence, the least value is 4 .

Solutions · 1