Let [x]= the greatest integer less than or equal to x and let $f (x) = sin x + cos x$ . Then the most general solutions of $f (x) = [f (\frac{π}{10})]$ are

2 n π + \frac{π}{2}, n ϵ Z

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n π, n ϵ Z

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2 n π, n ϵ Z

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none of these

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Solution

The correct option is C none of these
$f (x) = sin x + cos x$
$\Rightarrow f (x) = \sqrt{2} sin (x + \frac{π}{4})$
Also , given $f (x) = [f (\frac{π}{10})]$
Here, $f (\frac{π}{10}) = \sqrt{2} sin (\frac{7 π}{20})$
$\Rightarrow [f (\frac{π}{10})] = 1$
$\Rightarrow sin (x + \frac{π}{4}) = \frac{1}{\sqrt{2}}$
The general solution is
$x + \frac{π}{4} = n π + {(- 1)}^{n} \frac{π}{4}$
$\Rightarrow x = n π + {(- 1)}^{n} \frac{π}{4} - \frac{π}{4}$

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