JEE Main 2024 (Online) 9th April Evening ShiftQuestion 1
Let the range of the function $f(x)=\frac{1}{2+\sin3x+\cos3x}$, $x\in\mathbb{R}$ be $[a,b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to
Functions
Solution
$\sin3x+\cos3x\in[-\sqrt2,\sqrt2]$, so $f\in\left[\frac1{2+\sqrt2},\frac1{2-\sqrt2}\right]$. $\alpha=\frac12\cdot\frac{4}{2}=1$, $\beta=\frac1{\sqrt{2}}$, so $\frac\alpha\beta=\sqrt2$.