$d\=\sqrt{{\left(x-1\right)}^2\+{\left(y-2\right)}^2\+{\left(z-3\right)}^2}$

but the objective function is taken as $f\=d^2$, which simplifies the algebra significantly.

$-5 x-4 y\+3 z\+11$

$\= 0$

$-7 x\+2 y-6 z\+9$

$\= 0$

$-5 \mathrm{\λ}-7 \mathrm{\μ}\+2 x-2$

$\= 0$

$-4 \mathrm{\λ}\+2 \mathrm{\μ}\+2 y-4$

$\= 0$

$3 \mathrm{\λ}-6 \mathrm{\μ}\+2 z-6$

$\= 0$

Initialize

Loading Student:-MultivariateCalculus

$f\={\left(x-1\right)}^2\+{\left(y-2\right)}^2\+{\left(z-3\right)}^2$$\stackrel{\text{assign}}{\to }$

$g\=5 x\+4 y-3 z-11$$\stackrel{\text{assign}}{\to }$

$h\=7 x-2 y\+6 z- 9$$\stackrel{\text{assign}}{\to }$

Tools≻Tasks≻Browse:

Calculus - Multivariate≻Optimization≻Lagrange Multiplier Method

Table 4.9.9(a)   The Lagrange Multiplier Method task template

Optimization Assistant

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$\left(0.77\,2.97\,1.59\right)$ 

Implement the Lagrange multiplier method via first principles

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

$F$

$-\left(5x\+4y-3z-11\right)\mathrm{\λ}-\left(7x-2y\+6z-9\right)\mathrm{\μ}\+{\left(x-1\right)}^2\+{\left(y-2\right)}^2\+{\left(z-3\right)}^2$

$\stackrel{\text{gradient}}{\to }$

$\stackrel{\text{to list}}{\to }$

$\left[-5x-4y\+3z\+11\,-7x\+2y-6z\+9\,-5\mathrm{\λ}-7\mathrm{\μ}\+2x-2\,-4\mathrm{\λ}\+2\mathrm{\μ}\+2y-4\,3\mathrm{\λ}-6\mathrm{\μ}\+2z-6\right]$

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

$\left\{\mathrm{\λ}\=\frac{86}{257}\,\mathrm{\μ}\=-\frac{78}{257}\,x\=\frac{199}{257}\,y\=\frac{764}{257}\,z\=\frac{408}{257}\right\}$

Define the two constraint planes

$g\&equals;0$ = $5x\&plus;4y-3z-11\&equals;0$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 2\; >>}$$\stackrel{\text{assign to a name}}{\to }$$P$

$h\&equals;0$ = $7x-2y\&plus;6z-9\&equals;0$$\stackrel{\text{make plane}}{\to }$$\mathrm{<<\; Plane\; 3\; >>}$$\stackrel{\text{assign to a name}}{\to }$$Q$

Obtain the line of intersection of the planes

$P\&comma;Q$ = $\mathrm{<<\; Plane\; 2\; >>}\&comma;\mathrm{<<\; Plane\; 3\; >>}$$\stackrel{\text{intersection}}{\to }$$\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{assign to a name}}{\to }$$L$$$

Obtain the distance from $\left(1\&comma;2\&comma;3\right)$ to line $L$ 

$\left[1\&comma;2\&comma;3\right]\&comma;L$ = $\left[1\&comma;2\&comma;3\right]\&comma;\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{distance}}{\to }$$\frac{1}{4369}\sqrt{13073}\sqrt{4369}$$\stackrel{\text{simplify}}{\&equals;}$$\frac{1}{257}\sqrt{769}\sqrt{257}$$$

Obtain the parametric equations for the line of intersection

Write the name of the line of intersection.
Context Panel: Evaluate and Display Inline

Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation≻parametric

Context Panel: Assign to a Name≻$R$ 

$L$ = $\mathrm{<<\; Line\; 1\; >>}$$\stackrel{\text{representation}}{\to }$$\left[x\&equals;\frac{6947}{4369}\&plus;18t\&comma;y\&equals;\frac{170}{257}-51t\&comma;z\&equals;-\frac{588}{4369}-38t\right]$$\stackrel{\text{assign to a name}}{\to }$$R$$$

Obtain $f\left(x\left(t\right)\&comma;y\left(t\right)\&comma;z\left(t\right)\right)$ and minimize via elementary calculus

$\genfrac{}{}{0ex}{}{f}{\phantom{x\&equals;a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{R}$

${\left(\frac{2578}{4369}\&plus;18t\right)}^2\&plus;{\left(-\frac{344}{257}-51t\right)}^2\&plus;{\left(-\frac{13695}{4369}-38t\right)}^2$

$\stackrel{\text{simplify}}{\&equals;}$

$\frac{52277}{4369}\&plus;396t\&plus;4369t^2$

$\stackrel{\text{differentiate w.r.t. t}}{\to }$

$396\&plus;8738t$

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

$\left\{t\&equals;-\frac{198}{4369}\right\}$

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

$T$

Evaluate line $L$ at the critical number found for $t$ 

$\genfrac{}{}{0ex}{}{L}{\phantom{x\&equals;a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{T}$ = $\left[\frac{199}{257}\&comma;\frac{764}{257}\&comma;\frac{408}{257}\right]$$$

Initialize

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

$f≔{\left(x-1\right)}^2\&plus;{\left(y-2\right)}^2\&plus;{\left(z-3\right)}^2\&colon;$

$g≔5 x\&plus;4 y-3 z-11\&colon;$

$h≔7 x-2 y\&plus;6 z- 9\&colon;$

Implement the Lagrange multiplier method

$\mathrm{LagrangeMultipliers}\left(f\&comma;\left[g\&comma;h\right]\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left[\frac{199}{257}\&comma;\frac{764}{257}\&comma;\frac{408}{257}\right]$

$\mathrm{LagrangeMultipliers}\left(f\&comma;\left[g\&comma;h\right]\&comma;\left[x\&comma;y\&comma;z\right]\&comma;\mathrm{output}\&equals;\mathrm{detailed}\right)$

$\left[x\&equals;\frac{199}{257}\&comma;y\&equals;\frac{764}{257}\&comma;z\&equals;\frac{408}{257}\&comma;{\mathrm{\&lambda;}}_1\&equals;\frac{86}{257}\&comma;{\mathrm{\&lambda;}}_2\&equals;-\frac{78}{257}\&comma;{\left(x-1\right)}^2\&plus;{\left(y-2\right)}^2\&plus;{\left(z-3\right)}^2\&equals;\frac{769}{257}\right]$

Implement the Lagrange multiplier method from first principles

$F≔f- \mathrm{\&lambda;} g-\mathrm{\&mu;} h\&colon;$

$\mathrm{solve}\left(\mathrm{Equate}\left(\mathrm{Gradient}\left(F\&comma;\left[x\&comma;y\&comma;z\&comma;\mathrm{\&lambda;}\&comma;\mathrm{\&mu;}\right]\right)\&comma;⟨0\&comma;0\&comma;0\&comma;0\&comma;0⟩\right)\&comma;\left\{x\&comma;y\&comma;z\&comma;\mathrm{\&lambda;}\&comma;\mathrm{\&mu;}\right\}\right)$

$\left\{\mathrm{\&lambda;}\&equals;\frac{86}{257}\&comma;\mathrm{\&mu;}\&equals;-\frac{78}{257}\&comma;x\&equals;\frac{199}{257}\&comma;y\&equals;\frac{764}{257}\&comma;z\&equals;\frac{408}{257}\right\}$