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http://mathworld.wolfram.com/LagrangeMultiplier.htmlIf $A+B+C=\pi$. Then Minimum value of $\cot^2(A)+\cot^2(B)+\cos^2(C)$ is
What have you tried?If $A+B+C=\pi$. Then Minimum value of $\cot^2(A)+\cot^2(B)+\cos^2(C)$ is
This is a different question from the one you asked in the original post. Please clarify what the question really is.Thanks Caption Black But I did not Understand What steps i do after that means after converting into 2 variable.
would you like to explain it to me
We Know that If [tex]A+B+C = \pi[/tex], Then [tex]\tan (A)+\tan(B)+\tan(C) = \tan(A).\tan(B).\tan(C)[/tex]
Which we can prove easily
[tex]A+B=\pi-C\Leftrightarrow \tan(A+B) = \tan (\pi-C) = -\tan (C)[/tex]
So [tex]\frac{\tan(A)+\tan(B)}{1-\tan(A).\tan(B)} = -\tan (C)[/tex]
So [tex]\tan (A)+\tan(B)+\tan(C) = \tan(A).\tan(B).\tan(C)[/tex]
Now Using [tex]\mathbb{A.M}\geq \mathbb{G.M}[/tex]
[tex]\frac{\tan (A)+\tan(B)+\tan(C) }{3}\geq \left(\tan(A).\tan(B).\tan(C)\right)^{\frac{1}{3}}[/tex]
[tex]\frac{\tan(A).\tan(B).\tan(C)}{3}\geq \left(\tan(A).\tan(B).\tan(C)\right)^{\frac{1}{3}}[/tex]
[tex]\left(\tan(A).\tan(B).\tan(C)\right)^3\geq 27 \left(\tan(A).\tan(B).\tan(C)\right)[/tex]
So [tex]\left(\tan(A).\tan(B).\tan(C)\right)\geq 3\sqrt{3}[/tex]
bcz [tex]\tan(A).\tan(B).\tan(C)> 0[/tex]
So [tex]\cot(A).\cot(B).\cot(C)\leq \frac{1}{3\sqrt{3}}[/tex]
Now again Using [tex]\mathbb{A.M}\geq \mathbb{G.M}[/tex]
[tex]\frac{\cot^2(A)+\cot^2(B)+\cot^2(C)}{3}\geq \left(\cot(A).\cot(B).\cot(C)\right)^{\frac{2}{3}}[/tex]
So [tex]\cot^2(A)+\cot^2(B)+\cot^2(C)\geq 1[/tex]
and equality hold when [tex]A=B=C=\frac{\pi}{3}[/tex]
By the way equality holds when \( \cot^2(A)=\cot^2(B)=\cot^2(C) \) which does not imply that \(A=B=C \) unless you place some restriction on the allowed values of \(A,B\) and \(C\), which you have not done.So [tex]\cot^2(A)+\cot^2(B)+\cot^2(C)\geq 1[/tex]
and equality hold when [tex]A=B=C=\frac{\pi}{3}[/tex]
Given a function \(f(x,y,z)\) for which you seek a minimum subject to a constraint \(z=g(x,y)\), we look for the unconstrained minimum of \(h(x,y)=f(x,y,g(x,y))\).Thanks Caption Black But I did not Understand What steps i do after that means after converting into 2 variable.
would you like to explain it to me
Yes CaptionBlackBy the way equality holds when \( \cot^2(A)=\cot^2(B)=\cot^2(C) \) which does not imply that \(A=B=C \) unless you place some restriction on the allowed values of \(A,B\) and \(C\), which you have not done.
CB