Examples

Lagrange multipliers

Lagrange multipliers

Problem: Find the extreme values of $f(x,y,z)=3x+2y+4z$ on the ellipsoid $x^2+2y^2+6z^2=1$.
Solution: We define $g(x,y,z)=x^2+2y^2+6z^2$. Write out the Lagrange Equations:

$$\nabla f=(3,2,4),\nabla g=(2x,4y,12z),(3,2,4)=\lambda (2x,4y,12z).$$

Solve the system of equations by first solving for $\lambda$ in each coordinate and then equating these values:

$$\frac{3}{2}=\lambda x,\frac{1}{2}=\lambda y,\frac{1}{3}=\lambda z.$$

Obtain $x=\frac{9}{2}z$ and $y=\frac{3}{2}z$. Insert these expressions into the equation $g(x,y,z)=1$ and find a single equation, the solutions of which are $z_p=\frac{2}{\sqrt{123}}$ and $z_q=\frac{-2}{\sqrt{123}}$. Using these expressions again, we find the other coordinates of two critical points:

$$p=\frac{1}{\sqrt{123}}(9,3,2),q=-\frac{1}{\sqrt{123}}(9,3,2).$$

Plug these points into $f(x,y,z)$ and obtain $f(p)=\sqrt{\frac{41}{3}}$ and $f(q)=-\sqrt{\frac{41}{3}}$.

Since the ellipsoid has no boundary, these are the global max and min of $f$ on the ellipsoid.