Consider an acute angled triangle $\Delta xyz$. Suppose we want to minimise the function $f(x,y,z)=\tan x+\tan y+\tan z$. So as per the method of Lagrange multipliers here is what I did.
$$\begin{aligned}&\text{minimise}: f(x,y,z)=\tan x +\tan y+\tan z \\ &\text{given: }x+y+z=\pi \end{aligned}$$
$$\begin{aligned}\sec^2 x&=\lambda\\ \sec^2 y&=\lambda \\ \sec^2 z&=\lambda\end{aligned}$$
Now we have the constraint $x+y+z=\pi$ and $0\lt x,y,z\lt \pi/2$. Also do have $4$ unknowns and $4$ equations to solve for $x,y,z,\lambda$. But I am not clear on how to use these equations, managing the $\pm$ signs when we take square roots. Besides if it was not given that $\Delta xyz$ is acute angled, then the minimum would have been in the case of a degenerate triangle when any one of the angles is $0$ and others add up to $\pi$, in which case, $f(x,\pi-x,0)=0$, but I do not understand the jump from $0$ to $3\sqrt{3}$ either for the minimum. Can it not attain values in $(0,3\sqrt{3})$? Why? . Any hints are appreciated. Thanks.
Best Answer
To handle this problem with the Lagrange Multipliers method, we need first of all, convert the inequalities into equations so with the help of slack variables $(s_x,s_y,s_z,e_x,e_y,e_z)$ the formulation could be
$$ \min_{x,y,z} \tan x + \tan y + \tan z\ \ \ \text{s. t.}\cases{x+y+z=\pi\\ x = s_x^2\\ y = s_y^2\\ z = s_z^2\\ \frac{\pi}{2}-x = e_x^2\\ \frac{\pi}{2}-y = e_y^2\\ \frac{\pi}{2}-x = e_z^2\\ } $$
and the corresponding lagrangian
$$ L = \tan x + \tan y + \tan z + \lambda(x+y+z-\pi)+\sum_{k=\{x,y,z\}} \eta_k(k-s_k^2)+ \sum_{k=\{x,y,z\}} \mu_k\left(\frac{\pi}{2}-k-e_k^2\right) $$
so the set of stationary points for $L$ could contain the solution. Now the minimization problem has the variables $\{x,y,z,\lambda,\eta_k,\mu_k,s_k,e_k\},\ \ k=\{x,y,z\}$. Also, to avoid algebraic problems to obtain the solutions for equations involving trigonometric functions, we can use
$$ \tan x\approx x + \frac{x^3}{3} $$