Fools problem of the day,

$(1) $ Find the minimum value of $\tan^2 A +\tan^2 B+\tan^2 C$, given that $\tan A +\tan B+\tan C = 4 $

**NOTE**: Please don't post the solution here, instead use this thread for clarification of problem statement discussing strategy and checking the answer. The reason for this is once a solution is posted it act as a spoiler for others. At-least wait for a day before posting a solution. And If you are really confidant about your solution send my via private message.

Good luck!

Regards,
Fool/FFM!

Question

Fools problem of the day,

$(1) $ Find the minimum value of $\tan^2 A +\tan^2 B+\tan^2 C$, given that $\tan A +\tan B+\tan C = 4 $

**NOTE**: Please don't post the solution here, instead use this thread for clarification of problem statement discussing strategy and checking the answer. The reason for this is once a solution is posted it act as a spoiler for others. At-least wait for a day before posting a solution. And If you are really confidant about your solution send my via private message.

Good luck!

Regards,
Fool/FFM!

asnaseer · Answer

Can we assume A, B and C are the three angles within a triangle?

anonymous · Answer

No.

asnaseer · Answer

hmmm - if they were then I got an answer of 33.655
I guess it's back to the drawing board then...

zarkon · Answer

c=C?

anonymous · Answer

Yes, apologies for the typo.

zarkon · Answer

16/3?

anonymous · Answer

Yes, how do you do it? @Zarkon 

My approach involves some 3D geometry, so I was wondering for alternative approaches.

zarkon · Answer

$$x=\tan(A), y=\tan(B),z=\tan(C)$$

equations become 

$$x^2+y^2+z^2$$

with the constraint $x+y+z=4$

use Lagrange multipliers.

anonymous · Answer

What are Lagrange multipliers?

zarkon · Answer

it is a technique used to find max/min's of functions subject to constraints

zarkon · Answer

http://en.wikipedia.org/wiki/Lagrange_multiplier