If largest subset of $(0, π)$ at each point of which the function $f (x) = 3 {cos}^{4} x + 10 {cos}^{3} x + 6 {cos}^{2} x - 3$ is decreasing is $(0, \frac{π}{p}) \cup (\frac{2 π}{r}, π)$ , then find the value of $(p + r)$

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Solution

$f (x) = 12 {cos}^{2} x (- sin x) + 30 {cos}^{2} x (- sin x) + 12 cos x (- sin x)$
$= - 3 sin 2 x (2 {cos}^{2} x + 5 cos x + 2)$
$= - 3 sin 2 x (2 cos x + 1) (cos x + 2)$
$f (x) = 0 \Rightarrow sin 2 x = 0 \Rightarrow x = 0, \frac{π}{2}, π$
$2 cos x + 1 = 0$ , $\Rightarrow x = \frac{2 π}{3}$
If $f (x)$ decreases on $(0, \frac{π}{2}) \cup (\frac{2 π}{3} . π)$
$f (x)$ , $(0, \frac{π}{2}) \cup (\frac{2 π}{3}, π)$
$f (x) (0, \frac{π}{2}) \cup (\frac{2 π}{3}, π)$
and increases on $(\frac{π}{2}, \frac{2 π}{3}) o r (\frac{π}{2}, \frac{2 π}{3})$

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(0, π)

f (x) = 3 {cos}^{4} x + 10 {cos}^{3} x + 6 {cos}^{2} x - 3

(0, \frac{π}{p}) \cup (\frac{2 π}{r}, π)

(p + r)

(0, p)

f (x) = 3 {cos}^{4} x + 10 {cos}^{3} x + 6 {cos}^{2} x - 3

(0, \frac{π}{p}) \cup (\frac{2 π}{r}, π)

(p + r)

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1 + | tan x |)^{\frac{p}{| tan x |}} & , - \frac{π}{3} < x < 0 q & x = 0 e^{\frac{sin 3 x}{sin 2 x}}, & 0 < x < \frac{π}{3} \end{matrix}

x = 0

2 f (s i n x) + f (c o s x) = x \forall x ϵ

f (x) = ln sin x \forall x \in (0, \frac{π}{2}]

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