First slide

Trigonometric equations

Question

$I f \sqrt{3} \sin π x + \cos π x = x^{2} - \frac{2}{3} x + \frac{19}{9}, t h e n x i s e q u a l t o$

Moderate

A

$\frac{- 1}{3}$
B

$\frac{1}{3}$
C

$\frac{2}{3}$
D

$\frac{4}{3}$

Solution

$W e h a v e \sqrt{3} \sin π x + \cos π x = x^{2} - \frac{2}{3} x + \frac{19}{9} = {(x - \frac{1}{3})}^{2} + 2$
$N o w - 2 \leq \sqrt{3} \sin π x + \cos π x \leq 2, i . e ., L H S \leq 2, a n d R H S \geq 2 . ∴ L H S = R H S = 2 \Rightarrow x = \frac{1}{3}$

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