# How do you minimize and maximize f(x,y)=(x-y)(x+y)+sqrt(xy) constrained to 0<xy-y^2<5?

Jun 5, 2016

The points $\left\{x = \pm 5.15905 , y = \pm 3.86559\right\}$ are local minima

#### Explanation:

We will searching for stationary points, qualifying then as local maxima/minima.

First we will transform the maxima/minima with inequality restrictions into an equivalent maxima/minima problem but now with equality restrictions.

To do that we will introduce the so called slack variables ${s}_{1}$ and ${s}_{2}$ such that the problem will read.

Maximize/minimize $f \left(x , y\right) = \left(x - y\right) \left(x + y\right) + \sqrt{x y}$
constrained to

{ (g_1(x,y,s_1)=x y - y^2 - s_1^2=0), (g_2(x,y,s_2)=x y - y^2 + s_2^2 - 5=0) :}

The lagrangian is given by

$L \left(x , y , {s}_{1} , {s}_{2} , {\lambda}_{1} , {\lambda}_{2}\right) = f \left(x , y\right) + {\lambda}_{1} {g}_{1} \left(x , y , {s}_{1}\right) + {\lambda}_{2} {g}_{2} \left(x , y , {s}_{2}\right)$

The condition for stationary points is

$\nabla L \left(x , y , {s}_{1} , {s}_{2} , {\lambda}_{1} , {\lambda}_{2}\right) = \vec{0}$

so we get the conditions

{ (2 x + lambda_1 y + lambda_2 y + y/(2 sqrt[x y]) = 0), (lambda_1 (x - 2 y) + llambda_2 (x - 2 y) - 2 y + x/(2 sqrt[x y]) = 0), (-s_1^2 + x y - y^2 = 0), (-2 lambda_1 s_1 = 0), (-5 + s2^2 + x y - y^2 = 0), (2 lambda_2 s_2 = 0) :}

Solving for $\left\{x , y , {s}_{1} , {s}_{2} , {\lambda}_{1} , {\lambda}_{2}\right\}$ we have

{(x= -5.15905, y= -3.86559, lambda_1 = 0., s_1 = -2.23607, lambda_2= -2.78118, s_2= 0.), (x = 5.15905, y = 3.86559, lambda_1= 0., s_1 = -2.23607, lambda_2= -2.78118, s_2= 0.) :}

The restriction ${g}_{2} \left(x , y\right)$ is active (${\lambda}_{2} \ne 0$ and ${s}_{2} = 0$) so for qualifying the maxima/minima we produce

f_{g_2}(x) = 1/2 (10 + sqrt[2] sqrt[x (x - sqrt[-20 + x^2])] pm x (x + sqrt[x^2-20]))

Computing

$\frac{d}{\mathrm{dx}} \left({f}_{{g}_{2}} \left(\pm 5.15905\right)\right) = 0$

and

${d}^{2} / \left({\mathrm{dx}}^{2}\right) \left({f}_{{g}_{2}} \left(\pm 5.15905\right)\right) = 1.64412$

we conclude that the found solutions are local minima points.