The most general values of x for which $sin x + cos x = m i n {1, a^{2} - 4 a + 6}$ are given by

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None of these

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Solution

The correct option is C
$a^{2} - 4 a + 6 = (a - 2)^{2} + 2 > 1$ for all a
So, min ${1, a^{2} - 4 a + 6} = 1$
$∴$ The equation is $sin x + cos x = 1.$
$\Rightarrow sin (x + \frac{π}{4}) = \frac{1}{\sqrt{2}} = sin (\frac{π}{4})$
$\Rightarrow x + \frac{π}{4} = n π + (- 1)^{n} \frac{π}{4}$ or $x = 1 n π + (- 1)^{n} \frac{π}{4} - \frac{π}{4}$

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