Sequences & Series
Composition of Rational Function — Minimum Value
nta_pyq_2023_apr
Grade 11
Question:
If $f(x)=\dfrac{(\tan1°)x+\ln123}{x\ln1234-(\tan1°)}$, $x>0$, then $f(f(x))+f\!\left(f\!\left(\dfrac{4}{x}\right)\right)$ has minimum value
Step-by-Step Solution
Key Concept: Show $f(f(x))=x$ (involutory). Then $f(f(x))+f(f(4/x))=x+4/x\geq2\sqrt{4}=4$... wait: $x+4/x\geq 2\sqrt{x\cdot4/x}=4$? No: $x+4/x\geq2\sqrt{4}=4$. But answer is (3)=4.
Min of $x+\frac{4}{x}=4$ at $x=2$.
Correct Answer: 3