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1.Relation and Function
hard
If $f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{\varepsilon}(123)}{x \log _{\varepsilon}(1234)-\left(\tan 1^{\circ}\right)}, x > 0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is $...........$.
A
$8$
B
$4$
C
$2$
D
$0$
(JEE MAIN-2023)
Solution
Let $f(x)=\frac{A x+B}{C x-A}$
$f(f(x))=\frac{A\left(\frac{A x+B}{C x-A}\right)+B}{C\left(\frac{A x+B}{C x-A}\right)-A}=x$
$f\left(f\left(\frac{4}{x}\right)\right)=\frac{4}{x}f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)=x+\frac{4}{x} \geq 4(b y A M . \geq G . M .)$
Standard 12
Mathematics
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