5. Berilgen $$ f(x) $$ funkciyanı́ eń kishi oń periodın anıqlań: (5 ball) $$ f(x)=\sin 6 x+\cos \frac{4 x}{3}+\frac{1}{2} \operatorname{tg} \frac{6 x}{5} $$

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1. Har bir trigonometric funksiya uchun asosiy periodni aniqlaymiz:

- $$\sin 6x$$ funksiyasining periodi:
     $$
     T_1 = \frac{2\pi}{6} = \frac{\pi}{3}
     $$
     
   - $$\cos \frac{4x}{3}$$ funksiyasining periodi:
     $$
     T_2 = \frac{2\pi}{\frac{4}{3}} = \frac{2\pi \cdot 3}{4} = \frac{3\pi}{2}
     $$
     
   - $$\frac{1}{2}\tan \frac{6x}{5}$$ funksiyasining periodi:
     
     Tangens funksiyasining periodi $$\pi$$ ga teng bo‘lganligi sababli, argumenti $$\frac{6x}{5}$$ bo‘lgan holda:
     $$
     T_3 = \pi \cdot \frac{5}{6} = \frac{5\pi}{6}
     $$

2. Endi umumiy funksiya $$ f(x) $$ ning periodi, ya'ni eng kichik o‘ng period $$ T $$ ni topish uchun $$ T $$ ni quyidagi shartlarga mos bo‘lishi lozim:
   $$
   \frac{T}{T_1}\in\mathbb{Z},\quad \frac{T}{T_2}\in\mathbb{Z},\quad \frac{T}{T_3}\in\mathbb{Z}
   $$
   ya'ni
   $$
   \frac{T}{\frac{\pi}{3}},\quad \frac{T}{\frac{3\pi}{2}},\quad \frac{T}{\frac{5\pi}{6}} \text{ butun son bo‘lishi lozim.}
   $$

3. $$ T $$ ni $$\pi$$ ko‘paytuvchisi sifatida ifodalash uchun $$T=k\pi$$ deb yozamiz. Shunda har bir shart:
   
   - Birinchi shart:
     $$
     \frac{k\pi}{\pi/3} = 3k \quad \text{butun, har doim butun.}
     $$
   
   - Ikkinchi shart:
     $$
     \frac{k\pi}{3\pi/2} = \frac{2k}{3} \quad \text{butun bo‘lishi uchun } 2k \text{ 3 ga bo‘linishi kerak, ya'ni } k \text{ kamida 3ning ko‘paytmasi bo‘lishi lozim.}
     $$
   
   - Uchinchi shart:
     $$
     \frac{k\pi}{5\pi/6} = \frac{6k}{5} \quad \text{butun bo‘lishi uchun } 6k \text{ 5 ga bo‘linishi kerak, ya'ni } k \text{ kamida 5ning ko‘paytmasi bo‘lishi lozim.}
     $$

4. $$ k $$ eng kichik ijobiy butun son bo‘lishi uchun, u 3 va 5 ning EKOKi – ya'ni 15 ga teng bo‘ladi. Shunday qilib:
   $$
   T = 15\pi
   $$

5. Natijalarni tekshirib chiqamiz:
   
   - $$\sin 6x$$ uchun:
     $$
     \frac{15\pi}{\pi/3} = 45 \quad (\text{butun})
     $$
     
   - $$\cos \frac{4x}{3}$$ uchun:
     $$
     \frac{15\pi}{3\pi/2} = 10 \quad (\text{butun})
     $$
     
   - $$\tan \frac{6x}{5}$$ uchun:
     $$
     \frac{15\pi}{5\pi/6} = 18 \quad (\text{butun})
     $$

Shu sababli, berilgan $$ f(x) $$ funksiyasining eng kichik o‘ng periodi:
$$
\boxed{15\pi}
$$
Berilgan $$ f(x) $$ funksiyasining eng kichik o‘ng periodi $$ 15\pi $$ ga teng.

5. Berilgen \( f(x) \) funkciyanı́ eń kishi oń periodın anıqlań: (5 ball) \[ f(x)=\sin 6 x+\cos \frac{4 x}{3}+\frac{1}{2} \operatorname{tg} \frac{6 x}{5} \]

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