$\left(4 x-6 y\+5\right)\mathrm{\λ}\+3x\+4y-4\=0$

$\left(4y-4 x-4\right)\mathrm{\λ}\+6x\+3y\+5\=0$

$2 x^2-4 x y\+3 y^2\+4 x-5 y-1\=0$

$\left(-0.06694753468\,-0.2319153335\,0.7605175013\right)$

$\left(-1.056335968\,1.136359769\,-7.498087925\right)$

$\left(-2.609293059\,-1.213832530\,24.68260883\right)$

$\left(1.679706471\,2.119055768\,28.04552051\right)$

Initialize

$f\=3x^2\+3xy\+2y^2\+5x-4 y$$\stackrel{\text{assign}}{\to }$

$g\=2x^2-4xy\+3y^2\+4x-5 y-1.0$$\stackrel{\text{assign}}{\to }$

Obtain any unconstrained extrema

$\frac{\partial }{\partial x} f\=0\,\frac{\partial }{\partial y} f\=0$

$6x\+3y\+5\=0\,3x\+4y-4\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=-\frac{32}{15}\,y\=\frac{13}{5}\right\}$

Apply the Lagrange multiplier method via the  task template

The Lagrange multiplier method from first principles

$F\=f-\mathrm{\λ} g$$\stackrel{\text{assign}}{\to }$

$\frac{\partial }{\partial x} F\=0\,\frac{\partial }{\partial y} F\=0\,\frac{\partial }{\partial \mathrm{\λ}} F\=0$

$-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0$

$\stackrel{\text{assign to a name}}{\to }$

$Q$

$Q$

$-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{\mathrm{\λ}\=0.8374842625\,x\=-0.06694753468\,y\=-0.2319153335\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$s_1$

$Q$

$-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{\mathrm{\λ}\=9.558271317\,x\=1.679706471\,y\=2.119055768\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$s_2$

$Q$

$-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{\mathrm{\λ}\=-0.4341139647\,x\=-1.056335968\,y\=1.136359769\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$s_3$

$Q$

$-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{\mathrm{\λ}\=9.038358386\,x\=-2.609293059\,y\=-1.213832530\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$s_4$

Display the four solutions in the form $\left[x\,y\,f\left(x\,y\right)\right]$ 

$\genfrac{}{}{0ex}{}{\left[x\,y\,f\right]}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{s_1}$ = $\left[-0.06694753468\,-0.2319153335\,0.7605175013\right]$$$

$\genfrac{}{}{0ex}{}{\left[x\,y\,f\right]}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{s_2}$ = $\left[1.679706471\,2.119055768\,28.04552051\right]$$$

$\genfrac{}{}{0ex}{}{\left[x\,y\,f\right]}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{s_3}$ = $\left[-1.056335968\,1.136359769\,-7.498087925\right]$$$

$\genfrac{}{}{0ex}{}{\left[x\,y\,f\right]}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{s_4}$ = $\left[-2.609293059\,-1.213832530\,24.68260883\right]$$$

Numeric solution via the

The Optimization Assistant is no longer listed in Tools≻Assistants. It is now found in Tools≻Tutors≻Optimization

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$f≔3 x^2\+3xy\+2y^2\+5x-4 y\:$

$g≔2x^2-4xy\+3y^2\+4x-5 y-1.0\:$

Obtain any unconstrained extrema

$\mathrm{solve}\left(\left\{\mathrm{diff}\left(f\,x\right)\=0\,\mathrm{diff}\left(f\,y\right)\=0\right\}\right)$

$\left\{x\=-\frac{32}{15}\,y\=\frac{13}{5}\right\}$

Use the Lagrange multiplier method to search along the constraint ellipse

$$\begin{gathered} S≔\mathrm{LagrangeMultipliers}\left(f\,\left[g\right]\,\left[x\,y\right]\,\mathrm{output}\=\mathrm{detailed}\right)\: \\ \mathrm{print}~\left(\left[S\right]\right)\left[\right] \end{gathered}$$

$\left[x\=1.679706471\,y\=2.119055768\,{\mathrm{\λ}}_1\=9.558271317\,3x^2\+3xy\+2y^2\+5x-4y\=28.04552051\right]$

Implement the Lagrange multiplier method from first principles

$F≔f-\mathrm{\λ} g\:$

$q≔\left\{\mathrm{diff}\left(F\,x\right)\=0\,\mathrm{diff}\left(F\,y\right)\=0\,\mathrm{diff}\left(F\,\mathrm{\λ}\right)\=0\right\}$

$\left\{-\left(-4x\+6y-5\right)\mathrm{\λ}\+3x\+4y-4\=0\,-\left(4x-4y\+4\right)\mathrm{\λ}\+6x\+3y\+5\=0\,-2x^2\+4xy-3y^2-4x\+5y\+1.0\=0\right\}$

$s_1≔\mathrm{fsolve}\left(q\,\left\{x\,y\,\mathrm{\λ}\right\}\,\mathrm{\λ}\=0..1\right)$

$\left\{\mathrm{\λ}\=0.8374842625\,x\=-0.06694753468\,y\=-0.2319153335\right\}$

$s_2≔\mathrm{fsolve}\left(q\,\left\{x\,y\,\mathrm{\λ}\right\}\,\mathrm{\λ}\=-1..0\right)$

$\left\{\mathrm{\λ}\=-0.4341139647\,x\=-1.056335968\,y\=1.136359769\right\}$

$s_3≔\mathrm{fsolve}\left(q\,\left\{x\,y\,\mathrm{\lambda }\right\}\,\mathrm{\λ}\=9..9.5\right)$

$\left\{\mathrm{\λ}\=9.038358386\,x\=-2.609293059\,y\=-1.213832530\right\}$

$s_4≔\mathrm{fsolve}\left(q\,\left\{x\,y\,\mathrm{\λ}\right\}\,\mathrm{\λ}\=9.5..10\right)$

$\left\{\mathrm{\λ}\=9.558271317\,x\=1.679706471\,y\=2.119055768\right\}$

Use a do-loop containing the eval command to evaluate $x\,y$, and $f$ at the four solutions just obtained.

$$\begin{gathered} \mathbf{for} k \mathbf{from} 1 \mathbf{to} 4 \mathbf{do} \\ \mathrm{eval}\left(\left[x\,y\,f\right]\,s_k\right)\; \\ \mathbf{end} \mathbf{do} \end{gathered}$$

$\left[-0.06694753468\,-0.2319153335\,0.7605175013\right]$

$\mathrm{Optimization}:-\mathrm{Maximize}\left(f\,\left\{g\=0\right\}\right)$

$\left[28.0455205089150184\,\left[x\=1.67970647057851\,y\=2.11905576750032\right]\right]$

$\mathrm{Optimization}:-\mathrm{Minimize}\left(f\,\left\{g\=0\right\}\right)$

$\left[0.760517501131491080\,\left[x\=-0.0669475447599069\,y\=-0.231915341172423\right]\right]$