91影视

Given the constraint:$g(x,y,z)=0$

And the point (x, y, z) is given on the plane.

Let d be the distance from origin to the point.

$$\begin{gathered} d=\sqrt{x^2+y^2+z^2}. \\ 鈭�d(x,y,z)=\left(\frac{鈭�d}{鈭�x}\right)i+\left(\frac{鈭�d}{鈭�y}\right)j+\left(\frac{鈭�d}{鈭�z}\right)k \\ 鈭�d(x,y,z)=\left(\frac{x}{\sqrt{x^2+y^2+z^2}}\right)i+\left(\frac{y}{\sqrt{x^2+y^2+z^2}}\right)j+\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)k \\ Andgiven,鈭�g(x,y,z)=0. \end{gathered}$$

So, by Lagrange's method:

$$\begin{gathered} 鈭�d=位鈭�g, \\ \left(\frac{x}{\sqrt{x^2+y^2+z^2}}\right)i+\left(\frac{y}{\sqrt{x^2+y^2+z^2}}\right)j+\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)k=位\left(0i+0j+0k\right), \\ Thisimplies, \\ \frac{x}{\sqrt{x^2+y^2+z^2}}=0,\frac{y}{\sqrt{x^2+y^2+z^2}}=0and\frac{z}{\sqrt{x^2+y^2+z^2}}=0 \\ Whichisequaltox=0,y=0andz=0. \\ Therefore,theminimumoccursat(0,0,0). \end{gathered}$$

Let D be the distance from origin to the point.

$$\begin{gathered} D=x^2+y^2+z^2. \\ 鈭�D(x,y,z)=\left(\frac{鈭�D}{鈭�x}\right)i+\left(\frac{鈭�D}{鈭�y}\right)j+\left(\frac{鈭�D}{鈭�z}\right)k \\ 鈭�D(x,y,z)=\left(2x\right)i+\left(2y\right)j+\left(2z\right)k \\ Andgiven,鈭�g(x,y,z)=0. \end{gathered}$$

So, by Lagrange's method:

$$\begin{gathered} 鈭�D=位鈭�g, \\ \left(2x\right)i+\left(2y\right)j+\left(2z\right)k=位\left(0i+0j+0k\right), \\ Thisimplies, \\ 2x=0,2y=0and2z=0 \\ Whichisequaltox=0,y=0andz=0. \\ Therefore,theminimumoccursat(0,0,0). \end{gathered}$$

Therefore, in both the cases the solution is the same.