Initialize

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

$5 x\+3 y\+2 z-7$$\stackrel{\text{assign to a name}}{\to }$$g$

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

Form and solve the equations of the Lagrange multiplier method

$\frac{\partial }{\partial x} F\=0\,\frac{\partial }{\partial y} F\=0\,\frac{\partial }{\partial z} F\=0\,g\=0$

$-5\mathrm{\λ}\+2x-2\=0\,-3\mathrm{\λ}\+2y\+4\=0\,-2\mathrm{\λ}\+2z-6\=0\,5x\+3y\+2z-7\=0$

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

$\left\{\mathrm{\λ}\=\frac{2}{19}\,x\=\frac{24}{19}\,y\=-\frac{35}{19}\,z\=\frac{59}{19}\right\}$

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

$S$

Obtain the minimum distance between the given point and the given plane

$\sqrt{\genfrac{}{}{0ex}{}{f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S}}$ = $\frac{1}{19}\sqrt{38}$$\stackrel{\text{at 5 digits}}{\to }$$0.32444$

$\sqrt{f}\,g\=0$

$\sqrt{{\left(x-1\right)}^2\+{\left(y\+2\right)}^2\+{\left(z-3\right)}^2}\,5x\+3y\+2z-7\=0$

$\stackrel{\text{minimize}}{\to }$

$\left[0.324442842261524644\,\left[x\=1.26315789473688\,y\=-1.84210526315806\,z\=3.10526315789488\right]\right]$

Initialize

$f≔{\left(x-1\right)}^2\+{\left(y\+2\right)}^2\+{\left(z-3\right)}^2\:$$$

$g≔5 x\+3 y\+2 z-7\:$

Implement the Lagrange multiplier technique via the LagrangeMultipliers command

$S≔\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{LagrangeMultipliers}\left(f\,\left[g\right]\,\left[x\,y\,z\right]\right)$

$\left[\frac{24}{19}\,-\frac{35}{19}\,\frac{59}{19}\right]$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{LagrangeMultipliers}\left(f\,\left[g\right]\,\left[x\,y\,z\right]\,\mathrm{output}\=\mathrm{detailed}\right)$

$\left[x\=\frac{24}{19}\,y\=-\frac{35}{19}\,z\=\frac{59}{19}\,{\mathrm{\λ}}_1\=\frac{2}{19}\,{\left(x-1\right)}^2\+{\left(y\+2\right)}^2\+{\left(z-3\right)}^2\=\frac{2}{19}\right]$

$\mathrm{eval}\left(f\,\mathrm{Equate}\left(\left[x\,y\,z\right]\,S\right)\right)$ = $\frac{2}{19}$$$

Find the minimum distance between the given point and the given plane

$\mathrm{evalf}\left(\sqrt{2\/19}\right)$ = $0.3244428423$$$