If 3sinπx+cosπx=x2−23x+199, then x is equal to

If $\sqrt{3} \sin π x + \cos π x = x^{2} - \frac{2}{3} x + \frac{19}{9}$ , then x is equal to

A
$- \frac{1}{3}$
B
$\frac{1}{3}$
C
$\frac{2}{3}$
D
$\frac{4}{3}$

Solution:

LHS $= \sqrt{3} \sin π x + \cos π x \leq 2$ and RHS = $x^{2} - \frac{2}{3} x + \frac{19}{9} = {(x - \frac{1}{3})}^{2} + 2 \geq 2$

$∴ L H S = R H S = 2 \Rightarrow x = \frac{1}{3}$

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