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Grade: 12th pass

                        

Please Solve this ques with proper reasons....?.........................

2 years ago

Answers : (1)

Aditya Gupta
2068 Points
							
the above question has been taken from tmh jee advanced. it is a tricky question and its hard to understand the solution as well. lemme help u out tho, so u can get the reason as well.
Jensens Inequality: 

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.

now, if we set h(x)=sinx, h’(x)=cosx or h”(x)= – sinx. it is given in question that x1, x2, …..xn lie in the interval (0,π). in ths interval, sinx is positive, implying h”(x)

so, h((x1+x2+...xn)/n)=sin(π/n) >= (sinx1+sinx2+....sinxn)/n

so, the greatest value of the sum turns out to be n*sin(π/n), which occurs when all xi’s are equal to each other.

2 years ago
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