Let f(x) = \frac{cos 9 x + cos 6 x}{2 cos 5 x - 1} \left(\right. cos 5 x \neq \frac{1}{2} \left.\right) be a function, then which of the following is/are CORRECT ?

Engineering

Mathematics

Maxima and Minima

Question

Let f(x) = $\frac{\cos 9 x + \cos 6 x}{2 \cos 5 x - 1} (\cos 5 x \neq \frac{1}{2})$ be a function, then which of the following is/are CORRECT ?

Maximum value of y = f(x) is 2

y = f(x) is a one-one function in $[\frac{- π}{2}, \frac{π}{2}]$

The equation f(x) = 1 has 3 distinct real roots in $[0, \frac{π}{2}]$

f (x) = 0 has two distinct real roots in $[0, \frac{π}{2}]$

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$f (x) = \frac{\cos 9 x + \cos 6 x}{2 \cos 5 x - 1} = \frac{2 \cos \frac{15}{2} xcos \frac{3 x}{2}}{1 - 4 \sin^{2} \frac{5}{2} x}$

$= \frac{2 \cos \frac{3 x}{2} \cos \frac{5 x}{2} (1 - 4 \sin^{2} \frac{5}{2} x)}{(1 - 4 \sin^{2} \frac{5}{2} x)}$

= cos 4x + cos x = $2 \cos \frac{3 x}{2} \cos \frac{5 x}{2}$

⇒ f(x) $\leq$ 2

f(x) = 1

⇒ cos x = 1 – cos 4x = 2 sin²2x

cos x = 8 sin²x cos²x

cos x = 0 OR

8cos³x – 8 cos x + 1 = 0

cos x $\in$ Î[–1, 1].