Range of the function , f (x) = cot ^ -1 ( _ 4/5 , ,(5 , x^2 , , … - Vidyadip

MCQ

Range of the function , $f (x) = cot ^{-1}$ $\left( {{{\log }_{4/5}}\,\,(5\,{x^2}\,\, - \,\,8\,x\,\, + \,\,4)\,} \right)$ is :

A
$(0 , \pi )$
✓
$\left[ {\frac{\pi }{4}\,\,,\,\,\pi } \right)$
C
$\left( {0\,\,,\,\,\frac{\pi }{4}} \right]$
D
$\left( {0\,\,,\,\,\frac{\pi }{2}} \right)$

✓

Answer

Correct option: B.

$\left[ {\frac{\pi }{4}\,\,,\,\,\pi } \right)$

b
Let $g(x)=\left(5 x^{2}-8 x+4\right)$

$=5\left(x^{2}-\frac{8}{5} x+\frac{4}{5}\right)$

$=\left(5\left(x-\frac{4}{5}\right)^{2}+\frac{4}{5}\right)$

Thus, $R_{g}=\left[\frac{4}{5}, \infty\right)$

$\frac{4}{5} \leq g(x)<\infty$

$\log _{5 / 4}\left(\frac{4}{5}\right) \leq \log _{5 / 4}(g(x))<\log _{5 / 4}(\infty)$

$-1 \leq \log _{5 / 4}(g(x))<\infty$

$\tan ^{-1}(-1) \leq \tan ^{-1}\left(\log _{5 / 4}(g(x))\right)<\tan ^{-1}(\infty)$

$-\frac{\pi}{4} \leq \tan ^{-1}\left(\log _{5 / 4}((g(x)))<\frac{\pi}{2}\right.$

$-\frac{\pi}{4} \leq f(x)<\frac{\pi}{2}$

Thus, $\quad R_{f}=\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)$

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