Hàm lồi và cực trị của biểu thức dạng xtanA+ytanB+ztanC

Posted by Nguyen Song Minh

Edited by Nguyen Trung Tuan

In this post I = (a,b)\subset \mathbb{R}.
Theorem. If f: I\to\mathbb{R} such that f''(x) > 0\forall x\in I then f(c)\geq (c - d)f'(d) + f(d). Equal hold iff c = d. Where c,d\in I.

Proof. Cases follows shall occur
a)If c = d, we have equal,
b)If c > d then \dfrac {f(c) - f(d)}{c - d} = f'(m) for some m\in (d,c), since f''(x) > 0\forall x\in I hence f'(m) > f'(d) and we have done!
c)If c < d similary!

Corollary 1. Let f_{i}: I\to\mathbb{R}(i = 1,2,...,n)  such that  f''_{i}(x) > 0\forall i = 1,2,...,n\forall x\in I. Let x_{1},x_{2},...,x_{n};y_{1},y_{2},...,y_{n}\in I such that x_{1} + x_{2} + ... + x_{n} = y_{1} + y_{2} + ... + y_{n} and f'_{i}(y_{i}) = f'_{j}(y_{j})\forall i,j = 1,2,...,n. Then we have f_{1}(x_{1}) + f_{2}(x_{2}) + ... + f_{n}(x_{n})\geq f_{1}(y_{1}) + f_{2}(y_{2}) + ... + f_{n}(y_{n}) and equal holds iff x_{i} = y_{i}\forall i = 1,2,...,n.

Proof. Because f_{i}(x_{i})\geq (x_{i} - y_{i})f'_{i}(y_{i}) + f_{i}(y_{i})\forall i = 1,2,...,n therefore we have done.

Corollary 2. Let acute triangles ABC and MNP , then we have
\cos^{2}{M}\tan{A} + \cos^{2}{N}\tan{B} + \cos^{2}{P}\tan{C}\geq\\ \frac {1}{2}(\sin{2M} + \sin{2N} + \sin{2P}).

Proof. Using Corollary 1 with three functions f_{M}(x) = \cos^{2}{M}\tan{x},f_{N}(x) = \cos^{2}{N}\tan{x},\\ f_{P}(x) = \cos^{2}{P}\tan{x},I = (0,\frac {\pi}{2}) and x_{1} = A,x_{2} = B,x_{3} = C;y_{1} = M,y_{2} = N,y_{3} = P.

Now I will use Corollary 2 to solution problem follows

Problem(VMEO). Find minimum of \tan{A} + 2\tan{B} + 5\tan{C} where A,B,C are three angles of an acute triangle ABC.

Solution. By Corollary 2 we need find the acute triangle MNP such that \dfrac {\cos^{2}{M}}{1} = \dfrac {\cos^{2}{N}}{2} = \dfrac {\cos^{2}{P}}{5}. This is easy job, in fact, denote common value of fractions above is k then by \cos^{2}{M} + \cos^{2}{N} + \cos^{2}{P} + 2\cos{M}\cos{N}\cos{P} = 1 we have k = \frac {1}{10}.

Therefore \tan{A} + 2\tan{B} + 5\tan{C}\geq \\ 10.\frac {1}{2}(\sin{2M} + \sin{2N} + \sin{2P}) = 12.

P/S: We can find minimum of x\tan{A} + y\tan{B} + z\tan{C} by using Corollary2.

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