Question

if A+B+C=pi, then find the minimum value of cot^2 A+cot^2 B+cot^2 C

Answer 1

Lagrange's method? did you try it?

Answer 2

take A = B = C = pi/3

Answer 3

Remember the inequality\[a^2+b^2+c^2\ge ab+ac+bc\]which comes from\[(a-b)^2+(a-c)^2+(b-c)^2\ge0\]so i can write\[\cot^2 A+\cot^2 B+\cot^2 C\ge \cot A\cot B+\cot A\cot C +\cot B\cot C \]equality occurs when\[\cot A=\cot B=\cot C\]or\[A=B=C=\frac{\pi}{3}\]

Answer 4

Okay, but choosing the value of A,B,C also changes the value of the Left Hand Side, and Im not sure this will suffice to prove that this is a maximum.

Answer 5

pls....can u elaborate i m not getting it