Question

Find the maximum and minimum values of $si{n}^{-1}x+ta{n}^{-1}x.$

Answer 1

Let y $=si{n}^{-1}x+ta{n}^{-1}x$

Domain of y is y is $xϵ[-1,1]$

both $si{n}^{-1}x$ and $ta{n}^{-1}x$ are increasing

functions. So there minimum value

will occur for minimum value of x

and maximum value for maximum of x.

$\Rightarrow$ y minimum $=si{n}^{-1}(-1)+ta{n}^{-1}(-1)$

$=\frac{-\pi }{2}-\frac{\pi }{4}=\frac{-3\pi }{4}$

y (maximum) $=si{n}^{-1}\left(1\right)+co{s}^{-1}\left(1\right)$

$=\frac{\pi }{2}+\frac{\pi }{4}=\frac{3\pi }{4}$

$\Rightarrow$ minimum value $=\frac{-3\pi }{4}$

maximum value $=\frac{3\pi }{4}$