For the functions f( ) = ^2 + ^2 , and g( ) = ^2 + ^2 , > > 0 , let _ 0 < < /2 f( ) = _ 0 < < g( ) . If the first term of a G.P. is ( 2 ) , its common ratio is ( 2 ) and the sum of its first 10 terms is m n , (m, n) = 1 , then m + n is equal to _______.

Step-by-step solution

For the function f( ) = ^2 + ^2 , applying the AM-GM inequality gives: ^2 + ^2 2 ^2 ^2 = 2 Thus, _ 0 For the function g( ) = ^2 + ^2 , we can rewrite it as: g( ) = ^2 + (1 - ^2 ) = + ( - ) ^2 Since > > 0 , the maximum value of g( ) occurs when ^2 is maximum, which is 1 (at = /2 ). Thus, _ 0 Given that f( ) = g( ) , we have: 2 = Squaring both sides (since > 0 ), we get: 4 = ^2 = 4 For the given Geometric Progression (G.P.): First term a = 2 = 4 2 = 2 Common ratio r = 2 = 2 4 = 1 2 The sum of the first 10 terms is: S