find $\max$ and $\min$ with Lagrange multipliers. $f(x,y,z)=xyz^2$, $g(x,y,z)=x^2+y^2+z^2-1=0$

Question

Find the $\max$ and the $\min$ with Lagrange multipliers, given $$f(x,y,z)=xyz^2,$$ $$g(x,y,z)=x^2+y^2+z^2-1=0.$$

Answer 1

$$F(x,y,z)=(xyz^2)+ \lambda(x^2+y^2+z^2-1)$$ $$F_x=yz^2+2\lambda x$$ $$F_y=xz^2+2\lambda y$$ $$F_z=2xyz+2\lambda z$$ $$F_x,F_y,F_z=0$$ $$2xyz+2\lambda z=0$$ $$\lambda=-xy$$ $$xz^2-2xy^2=0$$ $$yz^2-2x^2y=0$$ $$z^2=2y^2$$ $$z^2=2x^2$$ $$z=\pm y\sqrt{2}$$ $$z=\pm x\sqrt{2}$$ $$y=\frac{\pm z}{\sqrt{2}}$$ $$x=\frac{\pm z}{\sqrt{2}}$$ Therefore, possible combinations of $x,y,z$are: $$\left(\frac{\pm z}{\sqrt{2}},\frac{\pm z}{\sqrt{2}}, z\right)$$

OR $$\left(\frac{ z}{\sqrt{2}},\frac{ z}{\sqrt{2}}, z\right),\left(\frac{ -z}{\sqrt{2}},\frac{ z}{\sqrt{2}}, z\right),\left(\frac{ z}{\sqrt{2}},\frac{ -z}{\sqrt{2}}, z\right),\left(\frac{- z}{\sqrt{2}},\frac{- z}{\sqrt{2}}, z\right)$$

Also, $$x^2+y^2+z^2=1$$ $$\frac{z^2}{2}+\frac{z^2}{2}+z^2=1$$ $$z^2=\frac{1}{2}$$ $$z=\frac{\pm 1}{\sqrt{2}}$$

Therefore, all possible values of $(x,y,z)$ are: $$\left(\frac{ 1}{{2}},\frac{ 1}{{2}}, \frac{1}{\sqrt{2}}\right),\left(\frac{ -1}{{2}},\frac{ 1}{{2}}, \frac{1}{\sqrt{2}}\right),\left(\frac{ 1}{{2}},\frac{ -1}{{2}}, \frac{1}{\sqrt{2}}\right),\left(\frac{ -1}{{2}},\frac{ -1}{{2}}, \frac{1}{\sqrt{2}}\right)$$ $$\left(\frac{ 1}{{2}},\frac{ 1}{{2}}, \frac{-1}{\sqrt{2}}\right),\left(\frac{ -1}{{2}},\frac{ 1}{{2}}, \frac{-1}{\sqrt{2}}\right),\left(\frac{ 1}{{2}},\frac{ -1}{{2}}, \frac{-1}{\sqrt{2}}\right),\left(\frac{ -1}{{2}},\frac{ -1}{{2}}, \frac{-1}{\sqrt{2}}\right)$$

The minimum possible value of $$xyz^2=\left(\frac{-1}{2}\right)\left(\frac{1}{2}\right)\left(\frac{1}{\sqrt{2}}\right)^2=\frac{-1}{8}$$

Answer 2

Using the technique of Lagrange multiplier is a very standard technique for such a problem. I would like to solve the problem as the following:

Note that $(x, y,z)$ lies on the unit sphere. Simply use the spherical coordinates, and then you have $$ f= \frac{1}{8} \sin (2 \theta) \sin ^2(2 \phi)$$. This gives us min and max of f.