First slide

Trigonometric equations

Question

If $a \cos x + b \cos 3 x \leq 1 \forall x \in R, t h e n |b|$

Moderate

A

is equal to 1
B

$\leq 1$
C

$\geq 1$
D

None of these

Solution

$G i v e n a \cos x + b \cos 3 x \leq 1$
$\begin{array}{l} F o r x = 0, a + b \leq 1 \to (i) \\ for x = \frac{π}{3}, \frac{a}{2} - b \leq 1 \to (i i) \\ for x = \frac{2 π}{3}, - \frac{a}{2} + b \leq 1 \to (i i i) \\ and for x = π, - a - b \leq 1 \to (i v) \\ F r o m (i) a n d (i i i) b \leq 1 a n d f r o m (i i) a n d (i v) - b \leq 1 \\ \Rightarrow - 1 \leq b \leq 1 \\ \Rightarrow |b| \leq 1 \end{array}$

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