91Ó°ÊÓ

Given function is$f(x,y,z)=x+y+z.$

Given constraint is$x^2+4y^2+16z^2=64.$

Gradients of function.

$$\begin{gathered} ∇f(x,y,z)=i+j+k \\ ∇g(x,y,z)=2xi+8yj+32zk \end{gathered}$$

Use the method of Lagrange multipliers.

$$\begin{gathered} ∇f(x,y,z)=\mathrm{λ}∇g(x,y,z) \\ i+j+k=\mathrm{λ}\left(2xi+8yj+32zk\right) \\ i+j+k=2\mathrm{λ}xi+8\mathrm{λ}yj+32\mathrm{λ}zk \end{gathered}$$

Compare terms.

$$\begin{gathered} 1=2\mathrm{λ}x⇒\mathrm{λ}=\frac{1}{2x} \\ 1=8\mathrm{λ}y⇒\mathrm{λ}=\frac{1}{8y} \\ 1=32\mathrm{λ}z⇒\mathrm{λ}=\frac{1}{32z} \\ \mathrm{so}x=4y\&z=\frac{y}{4} \end{gathered}$$

substitute $x=4y\&z=\frac{y}{4}$in constraint.

$$\begin{gathered} x^2+4y^2+16z^2=64 \\ {\left(4y\right)}^2+4y^2+16{\left(\frac{y}{4}\right)}^2=64 \\ 21y^2=64 \\ y=Â\pm \frac{8}{\sqrt{21}} \\ x=Â\pm \frac{32}{\sqrt{21}} \\ z=Â\pm \frac{2}{\sqrt{21}} \end{gathered}$$

so critical points are$\left(\frac{-32}{\sqrt{21}},\frac{-8}{\sqrt{21}},\frac{-2}{\sqrt{21}}\right)\&\left(\frac{32}{\sqrt{21}},\frac{8}{\sqrt{21}},\frac{2}{\sqrt{21}}\right).$

Find function value at $\left(\frac{-32}{\sqrt{21}},\frac{-8}{\sqrt{21}},\frac{-2}{\sqrt{21}}\right)\&\left(\frac{32}{\sqrt{21}},\frac{8}{\sqrt{21}},\frac{2}{\sqrt{21}}\right).$

role="math" localid="1649896217514" $$\begin{gathered} f(x,y,z)=x+y+z \\ f\left(\frac{-32}{\sqrt{21}},\frac{-8}{\sqrt{21}},\frac{-2}{\sqrt{21}}\right)=\left(\frac{-32}{\sqrt{21}}+\frac{-8}{\sqrt{21}}+\frac{-2}{\sqrt{21}}\right) \\ f\left(\frac{-32}{\sqrt{21}},\frac{-8}{\sqrt{21}},\frac{-2}{\sqrt{21}}\right)=\frac{-42}{\sqrt{21}} \\ f\left(\frac{-32}{\sqrt{21}},\frac{-8}{\sqrt{21}},\frac{-2}{\sqrt{21}}\right)=-2\sqrt{21} \\ \mathrm{and}f\left(\frac{32}{\sqrt{21}},\frac{8}{\sqrt{21}},\frac{2}{\sqrt{21}}\right)=2\sqrt{21} \end{gathered}$$

So the maximum value of the function is $2\sqrt{21}$and the minimum value is $-2\sqrt{21}.$

As constraint is bounded and closed ellipsoid so maximum and minimum both exist.