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Please Solve this ques with proper reasons....?.........................
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 is a twice differentiable, real-valued function on an interval and that for all . Then, for every positive integer m and for all points in , we haveMoreover, equality holds if and only if . A similar result holds if for all 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)/nso, 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.
Suppose is a twice differentiable, real-valued function on an interval and that for all . Then, for every positive integer m and for all points in , we have
Moreover, equality holds if and only if . A similar result holds if
for all 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.
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