If A and B are acute angles such that 2A+2B= , then what is the maximum value of A B ?

Step-by-step solution

Given 2A + 2B = , which simplifies to A + B = 2 . Since A and B are acute angles, B = 2 - A . The expression to maximize is A B . Substituting B = 2 - A , we get A ( 2 - A ) = A A . Multiplying and dividing by 2 , we obtain 2 A A 2 = 2A 2 . The maximum value of 2A is 1 , which occurs when 2A = 2 , or A = 4 . Therefore, the maximum value of A B is 1 2 1 = 1 2 . Answer: 1 2