If the domain of the function f(x)= ^-1((2x-5)/(11-3x))+ ^-1(2x^2-3x+1) is the interval [ …

Question

If the domain of the function $f(x)={\cos }^{-1}(\frac{2x-5}{11-3x})+{\sin }^{-1}(2{x}^{2}-3x+1)$ is the interval $[\alpha ,\beta ]$, then $\alpha +2\beta $ is equal to :

Accepted Answer

$\frac{2x-5}{11-3x}-1\le 0$

$\begin{gathered} \Rightarrow \frac{2x-5-11+3x}{11-3x}\le 0 \ \Rightarrow \frac{5x-16}{11-3x}\le 0 \ \Rightarrow x\in (-\infty,\frac{16}{5}]\cup (\frac{11}{3},\infty)\ldots \end{gathered}$

$\begin{gathered} (2) \frac{2x-5}{11-3x}\ge -1 \ \Rightarrow \frac{2x-5}{11-3x}+1\ge 0 \ \Rightarrow \frac{6-x}{11-3x}\ge 0 \ \Rightarrow \frac{x-6}{3x-11}\ge 0 \ \Rightarrow x\in (-\infty,\frac{11}{3})\cup [6,\infty)\ldots $

$\begin{gathered} (3)2{x}^{2}-3x+1\le 1 \ \Rightarrow \, x(2x-3)\le 0 \ x\in [0,\frac{3}{2}] \end{gathered}$

$\begin{gathered} (4)2{x}^{2}-3x+1\ge -1 \ \Rightarrow 2{x}^{2}-3x+2\ge 0 \ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, x\in R\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(iv) \end{gathered}$

$ \ x \in [0,\frac{3}{2}] \ 	herefore \, \alpha +2\beta \ =3 \end{gathered}$

Solutions · 1