Maximum and minimum of $x^2+2y^2+3z^2$ subject to $x^2+y^2+z^2=100$

Question

Find the maximum and minimum values of the function $f(x,y,z)=x^2+2y^2+3z^2$ subject to the constraint $x^2+y^2+z^2=100$.

I know to find the critical points I need to solve the system of equations

$$\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$$

I ended up with

$$\begin{aligned} 2x &= \lambda 2x\\ 4y &= \lambda 2y\\ 6z &= \lambda 2z\end{aligned}$$

I don't know to go from here since $\lambda$ ends up as $1$, $2$, and $3$.

Answer 1

You can find the maximum and the minimum by inspection, without using any theorem. Note that $x^{2}+2y^{2}+3z^{3}=100+y^{2}+2z^{2} \leq 100+(2y^{2}+2z^{2}) \leq 100+200=300$. The value $300$ is attained when $x=0,y=0$ and $z=10$.

Similarly a lower bound is $100$ and this bound is attained when $x=10,y=0,z=0$.

Answer 2

Lagrange function is: $$L=x^2+2y^2+3z^2+\lambda (100-x^2-y^2-z^2)\\ \begin{cases} L_x=2x-2x\lambda=0\\ L_y=4y-2y\lambda=0\\ L_z=6z-2z\lambda=0\\ L_\lambda=x^2+y^2+z^2=100 \end{cases} \Rightarrow \\ (x,y,z,\lambda)=(0,0,\pm10,3),(0,\pm10,0,2),(\pm10,0,0,1)$$ Now check the candidate (critical) points: $$f(0,0,\pm 10)=300 \ \max\\ f(0,\pm10,0)=200 \ \ \ \ \ \ \ \ \ \\ f(\pm10,0,0)=100 \ \min$$

Answer 3

This is a sort of "dual" of the extremization problem that is often asked here [for example: Lagrange Multipliers to find the maximum and minimum values ] , with the function being $ \ x^2 + y^2 + z^2 \ $ under the constraint $ \ ax^2 + by^2 + cz^2 \ = \ d \ $ with specified coefficients. The geometrical interpretation is similar, though. The constraint surface $ \ x^2 + y^2 + z^2 \ = \ 100 \ $ represents a spherical surface of radius $ \ 10 \ $ centered on the origin. Level surfaces of $ \ x^2 + 2y^2 + 3z^2 \ $ are triaxial ellipsoids of varying sizes, with all of them being similar.

For a particular ellipsoid $ \ x^2 + 2y^2 + 3z^2 \ = \ C \ \ \rightarrow \ \ \frac{x^2}{C} \ + \ \frac{y^2}{C/2} \ + \ \frac{z^2}{C/3} \ = \ 1 \ \ , $ the semi-axis lengths are $ \ \sqrt{C} \ , \ \sqrt{C/2} \ $ and $ \ \sqrt{C/3} \ $ . The smallest of this "family" of ellipsoids which will be tangent to the spherical surface is the one for which $ \ \sqrt{C} \ = \ 10 \ \Rightarrow \ C \ = \ 100 \ \ ; $ this gives us an ellipsoid inscribed within the sphere. The largest ellipsoid that can be tangent to the spherical surface is the one for which its shortest axis makes contact (the ellipsoid circumscribes the sphere); for this, $ \ \sqrt{C/3} \ = \ 10 \ \Rightarrow \ C \ = \ 300 \ \ . $ So the constrained minimum and maximum values of $ \ x^2 + 2y^2 + 3z^2 \ $ are $ \ 100 \ $ and $ \ 300 \ $ , respectively, as also shown in the other answers. [The ellipsoid which has the endpoints of its "middle-length" axis tangent to the spherical surface has $ \ \sqrt{C/2} \ = \ 10 \ \Rightarrow \ C \ = \ 200 \ \ , $ which has no special importance here.]