The problem involves finding the minimum value of the product of the tangents of three acute angles, given that their sum equals 180 degrees. Additionally, there is a second part that requires proving an inequality involving the sum of the tangents when the angles sum to 90 degrees.
Several participants have offered different approaches to tackle the first problem, including algebraic manipulations and geometric interpretations. There is a recognition of the need for more attempts to engage with the problems, particularly for the second question, but no consensus has been reached on a specific method.
Participants note that the angles are acute and that specific mathematical techniques, such as Lagrange multipliers, may be applicable. There is an emphasis on the importance of showing attempts in order to receive further assistance.
NeroKid said:you can use the langrange function to do it, it is quite easy if u use it
Michael_Light said:1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)
ehild said:Following rock.freak's suggestion, I would write the tangent of the third angle in terms of the tangents of the other two angles, and would use the relation between geometric and arithmetic means.
ehild
NeroKid said:tan(a)*tan(b)*tan(c) = -tan(b+c) *tan(b)*tan(c) = (tanb+tanc)/(1-tanb*tanc) *tanb*tanc
call S = tanb + tanc and P = tanb * tanc S^2>= 4P this should be easy from here on
Michael_Light said:Still cannot do...