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Please solve this question with explanation I am not able to solve these type of questions

Please solve this question with explanation
I am not able to solve these type of questions
 

Question Image
Grade:12th pass

1 Answers

Aditya Gupta
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<x<b. 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<x<b 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

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