![a)](/media/m/f/0/8/f0844437a160b45486aedcc02b92949d.png)
Ako su
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
,
![x_2\in (0,\displaystyle\frac{\pi }{2})](/media/m/2/f/f/2ff61717c94b9d70b5c2443d0046c8cc.png)
, dokažite:
![b)](/media/m/d/2/f/d2f292cd6a69e9158afe71ba9d830da4.png)
Ako su
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
,
![x_2](/media/m/a/a/1/aa16f4edacb7b534405242617406658f.png)
,
![\dots](/media/m/3/6/1/36118a223c1f6e75548277354fbabc8a.png)
,
![x_{2^k}\in \left(0,\displaystyle\frac{\pi}{2}\right)](/media/m/2/c/2/2c2e788d659beeeb66f3401d92dc9173.png)
, dokažite:
![c)](/media/m/4/6/b/46bcc39b0cc49ad861e814d96288b71d.png)
Ako su
![x_1](/media/m/9/2/a/92aefd356eeab9982f45f21fb206a2ef.png)
,
![x_2](/media/m/a/a/1/aa16f4edacb7b534405242617406658f.png)
,
![\dots](/media/m/3/6/1/36118a223c1f6e75548277354fbabc8a.png)
,
![x_n\in \left(0,\displaystyle\frac{\pi}{2}\right)](/media/m/7/5/e/75eced4fea164c649af8a0d5adafe8d0.png)
, dokažite:
%V0
$a)$ Ako su $x_1$, $x_2\in (0,\displaystyle\frac{\pi }{2})$, dokažite: $$
\frac{\cos x_1+\cos x_2}{2}\leq \cos \frac{x_1+x_2}{2}.
$$ $b)$ Ako su $x_1$, $x_2$, $\dots$, $x_{2^k}\in \left(0,\displaystyle\frac{\pi}{2}\right)$, dokažite: $$
\frac{1}{2^k} \sum _{j=1}^{2^k} \cos x_j \leq \cos \left( \dfrac{1}{2^k} \sum_{j=1}^{2^k} x_j \right), \,\, \text{za} \, k \in \mathbb{N} \text{.}
$$ $c)$ Ako su $x_1$, $x_2$, $\dots$, $x_n\in \left(0,\displaystyle\frac{\pi}{2}\right)$, dokažite: $$
\frac{1}{n} \sum _{j=1}^{n} \cos x_j \leq \cos \left( \dfrac{1}{n} \sum_{j=1}^{n}x_j \right), \,\, \text{za} \, n \in \mathbb{N} \text{.}
$$