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# Please solve this question with explanationI am not able to solve these type of questions

2075 Points
2 years ago
jensen inequality states that:

Suppose $h(x)$ is a twice differentiable, real-valued function on an interval $[a,b]$ and that $h^{}(x)>0$ for all $a. Then, for every positive integer m and for all points $x_{1}, x_{2}, \ldots x_{m}$ in $[a,b]$, we have

$h(\frac{x_{1}+x_{2}+\ldots+x_{m}}{m}) \leq \frac{h(x_{1})+h(x_{2})+h(x_{3})+\ldots+h(x_{m})}{m}$

Moreover, equality holds if and only if $x_{1}=x_{2}=\ldots=x_{m}$. A similar result holds if

$h^{}(x)<0$ for all $a except that the inequality sign is reversed.

take h(x)=sinx, so that h”(x)= – sinx

so, we get sin((x1+x2+....+xn)/n)=sin(pi/n)>=(sinx1+sinx2+.....+sinxn)/n

so, the greatest value of the sum turns out to be n*sin(pi/n), and it occurs when each xi=pi/n