1. Muammo: $4\cos^3 x + 3\sin x = 3\cos x + 4\sin^3 x$ tenglamaning $[0;2\pi]$ oraliqdagi eng katta va eng kichik ildizlari yig'indisini topish va $[-\pi;0]$ oraliqda nechta ildizga ega ekanligini aniqlash. 2. Tenglamani soddalashtirish uchun trigonometriya identifikatsiyalaridan foydalanamiz. E'tibor bering, $\cos^3 x$ va $\sin^3 x$ ni quyidagicha ifodalash mumkin: $$\cos^3 x = \cos x (1 - \sin^2 x), \quad \sin^3 x = \sin x (1 - \cos^2 x)$$ 3. Tenglamani chap va o'ng tomonlarini quyidagicha yozamiz: $$4\cos^3 x + 3\sin x = 4\cos x (1 - \sin^2 x) + 3\sin x$$ $$3\cos x + 4\sin^3 x = 3\cos x + 4\sin x (1 - \cos^2 x)$$ 4. Tenglama: $$4\cos x - 4\cos x \sin^2 x + 3\sin x = 3\cos x + 4\sin x - 4\sin x \cos^2 x$$ 5. Barcha terminlarni chapga olib kelamiz: $$4\cos x - 4\cos x \sin^2 x + 3\sin x - 3\cos x - 4\sin x + 4\sin x \cos^2 x = 0$$ 6. Soddalashtiramiz: $$(4\cos x - 3\cos x) + (3\sin x - 4\sin x) - 4\cos x \sin^2 x + 4\sin x \cos^2 x = 0$$ $$\cos x - \sin x - 4\cos x \sin^2 x + 4\sin x \cos^2 x = 0$$ 7. $\cos^2 x = 1 - \sin^2 x$, shuning uchun: $$\cos x - \sin x - 4\cos x \sin^2 x + 4\sin x (1 - \sin^2 x) = 0$$ 8. Tenglama: $$\cos x - \sin x - 4\cos x \sin^2 x + 4\sin x - 4\sin^3 x = 0$$ 9. Bu yerda $\cos x$ va $\sin x$ aralashgan, lekin tenglama simmetrik ko'rinishda. Boshqa yondashuv sifatida, $x$ ni $\alpha$ deb olamiz va $t = \tan x$ ni ko'rib chiqamiz, ammo $\tan x$ ni ishlatish uchun $\cos x \neq 0$ bo'lishi kerak. 10. Yana bir yondashuv: $\cos x = a$, $\sin x = b$ va $a^2 + b^2 = 1$. 11. Tenglama: $$4a^3 + 3b = 3a + 4b^3$$ $$4a^3 - 3a = 4b^3 - 3b$$ 12. E'tibor bering, $4t^3 - 3t = \cos 3\theta$ yoki $\sin 3\theta$ formulasiga o'xshash. Aslida: $$\cos 3x = 4\cos^3 x - 3\cos x$$ $$\sin 3x = 3\sin x - 4\sin^3 x$$ 13. Tenglamani qayta yozamiz: $$4a^3 - 3a = 4b^3 - 3b$$ $$\Rightarrow \cos 3x = 3\sin x - 4\sin^3 x$$ 14. Ammo $3\sin x - 4\sin^3 x = \sin 3x$, shuning uchun: $$\cos 3x = \sin 3x$$ 15. Bu tenglama: $$\cos 3x = \sin 3x$$ $$\Rightarrow \tan 3x = 1$$ 16. $\tan 3x = 1$ bo'lsa, $$3x = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}$$ 17. $x$ ni topamiz: $$x = \frac{\pi}{12} + \frac{k\pi}{3}$$ 18. Endi $[0; 2\pi]$ oraliqda ildizlarni topamiz: $$0 \leq x \leq 2\pi$$ $$0 \leq \frac{\pi}{12} + \frac{k\pi}{3} \leq 2\pi$$ 19. $k$ uchun chegaralarni topamiz: $$-\frac{\pi}{12} \leq \frac{k\pi}{3} \leq 2\pi - \frac{\pi}{12}$$ $$-\frac{1}{12} \leq \frac{k}{3} \leq 2 - \frac{1}{12} = \frac{23}{12}$$ $$-\frac{1}{4} \leq k \leq \frac{23}{4} = 5.75$$ 20. $k$ butun son bo'lgani uchun $k = 0,1,2,3,4,5$. 21. Ildizlar: $$x_k = \frac{\pi}{12} + \frac{k\pi}{3}, \quad k=0,1,2,3,4,5$$ 22. Eng kichik ildiz $k=0$ uchun: $$x_0 = \frac{\pi}{12}$$ 23. Eng katta ildiz $k=5$ uchun: $$x_5 = \frac{\pi}{12} + \frac{5\pi}{3} = \frac{\pi}{12} + \frac{20\pi}{12} = \frac{21\pi}{12} = \frac{7\pi}{4}$$ 24. Eng katta va eng kichik ildizlar yig'indisi: $$x_0 + x_5 = \frac{\pi}{12} + \frac{7\pi}{4} = \frac{\pi}{12} + \frac{21\pi}{12} = \frac{22\pi}{12} = \frac{11\pi}{6}$$ 25. $[-\pi;0]$ oraliqda ildizlar soni uchun: $$-\pi \leq x = \frac{\pi}{12} + \frac{k\pi}{3} \leq 0$$ 26. $k$ uchun chegaralar: $$-\pi - \frac{\pi}{12} \leq \frac{k\pi}{3} \leq -\frac{\pi}{12}$$ $$-\frac{13\pi}{12} \leq \frac{k\pi}{3} \leq -\frac{\pi}{12}$$ 27. $k$ ni topamiz: $$-\frac{13}{12} \leq \frac{k}{3} \leq -\frac{1}{12}$$ $$-\frac{39}{12} \leq k \leq -\frac{3}{12}$$ $$-3.25 \leq k \leq -0.25$$ 28. $k$ butun son bo'lgani uchun $k = -3, -2, -1$. 29. Demak, $[-\pi;0]$ oraliqda ildizlar soni $3$ ta. Javoblar: \begin{itemize} \item a) Eng katta va eng kichik ildizlar yig'indisi $\frac{11\pi}{6}$ \item b) $[-\pi;0]$ oraliqda ildizlar soni $3$ \end{itemize}