
If for and then the greatest value of the sum
A. n
B.
C. n sin
D. 0
Answer
518.4k+ views
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 curve 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
And as given in question,
and
for all
In the interval (0, π) and
So, the graph of is concave downward graph
For this question we are Applying Jensen's inequality for concave downward function we have
Since
since
Therefore, the greatest value of the sum is n sin .
Thus, option (C) is the correct answer.
Note- We need to remember, In the interval (0, π), the graph of is a concave downward graph. While In the interval (π, 2 π), the graph of 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.
Complete step-by-step solution -
We need to calculate,
greatest value of the sum
And as given in question,
In the interval (0, π) and
So, the graph of
For this question we are Applying Jensen's inequality for concave downward function we have
Since
since
Therefore, the greatest value of the sum
Thus, option (C) is the correct answer.
Note- We need to remember, In the interval (0, π), the graph of
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