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If xi>0,for 1inand x1+x2+.....+xn=π then the greatest value of the sumsinxi+sinx2+....+sinxn=......
A. n
B. π
C. n sin(πn)
D. 0

Answer
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HINT- We can solve this question by applying Jensen's inequality for concave downward function. Johan Jensen is the first person who described mathematical rule and the mathematical rule is also known generally as Jensen's Inequality. First, we identify the nature of the f(x)=sinxcurve and then Applying Jensen's inequality in the interval (0, π).

Complete step-by-step solution -
We need to calculate,
greatest value of the sum
sinxi+sinx2+....+sinxn
And as given in question,
 xi>0,1in and x1+x2+x3+....+xn=π
xiπ for all 1in
f(x)=sinx
f(x)=cosx
f(x)=sinx
In the interval (0, π) and f(x)<0
So, the graph of f(x)=sinxis concave downward graph
For this question we are Applying Jensen's inequality for concave downward function we have
f(x1+x2+x3+....xnn)f(x1)+f(x2)+f(x3)+......+f(xn)n
Since f(x)=sinr
sin(x1+x2+x3+....xnn) sinx1+sinx2+sinx3+......sinxnn
sin(πn)sinx1+sinx2+sinx3+......sinxnn
since x1+x2+x3+......+xn=π
sinx1+sinx2+sinx3+.......sinxnnsin(πn)
Therefore, the greatest value of the sumsinxi+sinx2+....+sinxn is n sin(πn).
Thus, option (C) is the correct answer.

Note- We need to remember, In the interval (0, π), the graph of f(x)=sinx is a concave downward graph. While In the interval (π, 2 π), the graph of f(x)=sinx is a concave upward graph. And also, in this question, students should be careful while considering all the cases which add up to n as missing out any change gives an overall change in the result. and also avoid basic calculations mistakes during calculations.

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