Question:
Let $\alpha\cot x+\beta\cot y+\gamma\cot z=\psi$ then the minimum value of $\cot^2x+\cot^2y+\cot^2z$ is
A) $\frac{\psi}{\alpha+\beta+\gamma}$
B) $\frac{\psi^2}{\alpha^2+\beta^2+\gamma^2}$
C) $\frac{|\psi|}{\sqrt{\alpha^2+\beta^2+\gamma^2}}$
D) $\psi\sqrt{\alpha^2+\beta^2+\gamma^2}$
My Attempt:
I first thought of weighted AM-GM inequality but couldn't conclude.
Then I squared the given equation and tried AM-GM inequality pairwise on the three terms, still couldn't conclude.
Then I took $\alpha=\beta=\gamma=1$ and $x=y=z=\frac\pi4$ then I got option B. Not sure if it is correct.
What's the proper way to solve it?
Best Answer
By C-S $$\sum_{cyc}\cot^2x\sum_{cyc}\alpha^2\geq\left(\sum_{cyc}\alpha\cot{x}\right)^2$$ and from here $$\sum_{cyc}\cot^2x\geq\frac{\psi^2}{\alpha^2+\beta^2+\gamma^2}.$$ The equality occurs, when $$\left(\cot{x},\cot{y},\cot{z}\right)||(\alpha,\beta,\gamma),$$ which says that we got a minimal value.