Find max and min of $x^3+y^3+z^3$ on the plane $x+y+z=4$ via Lagrange multipliers

Question

I have this easy question please:

Find the maximum and minimum values of the function $ (, , ) = ^3 + ^3 + ^3$ on the plane $ + + = 4$ by Lagrange multipliers method.

My attempt is shown in the picture. I reached to nine possiblitiest from which I need to substitute for $\lambda$ to find the min/max. But when I find $\lambda$, I got two posiiblilities with two values of $ (, , )$.

I wonder if my answer is correct or not.

Answer 1

No, it is not correct.

You have the system$$\left\{\begin{array}{l}3x^2=\lambda\\3y^2=\lambda\\3z^2=\lambda\\x+y+z-4=0.\end{array}\right.$$So, for some number $\lambda$ (which has to be greater than or equal to $0$), you have that each of the numbers $x$, $y$, and $z$ is equal to $\pm\sqrt{\frac\lambda3}$. Since they're all equal and their sum is greater than $0$, then either all of them are equal to $\sqrt\frac\lambda3$ or two of them are equal to $\sqrt\frac\lambda3$, whereas the third one is equal to $-\sqrt\frac\lambda3$.

If they are all equal to $\sqrt\frac\lambda3$, then$$4=x+y+z=3\sqrt\frac\lambda3=\sqrt{3\lambda},$$and therefore $\lambda=\frac{16}3$ and $x=y=z=\frac43$. Otherwise$$4=x+y+z=\sqrt{\frac\lambda3},$$and therefore $\lambda=48$ and$$(x,y,z)=(-4,4,4),\ (x,y,z)=(4,-4,4)\text{, or }(x,y,z)=(4,4,-4).$$Can you take it from here?

Answer 2

This is not intended as an answer, but it does not fit as a comment. The surface for $ \ x^3 + y^3 + z^3 = c > 0 \ $ has a three-fold symmetry about the line $ \ x = y = z \ $ , has a large "bump" in the first octant, and three symmetrical ridges which extend infinitely. So for the surface with $ \ x^3 + y^3 + z^3 = c_{min} \ \ , $ corresponding to $ \ \lambda = \frac{16}{3} \ \ , $ the "bump" just contacts the plane $ \ x + y + z = 4 \ $ at one point [graph at left below; $ \ c \ $ was set a little above that value so that the point would be visible]. We have a tangent point in this case: this is what the Lagrange method detects since the normals to the surface and plane are parallel there. But we can make $ \ c \ $ negative indefinitely and the surface ridges still intersect the plane farther and farther from the origin. Hence, there is no global minimum.

However, for the value of $ \ c \ $ for $ \lambda = 48 \ , $ the surface $ \ x^3 + y^3 + z^3 = c_{"max"} \ \ $ breaks through the plane in such a way that the three ridges have places where they narrow to points [graph at right below]. These are not tangent points, but again the normals to the surface and the plane are parallel at those locations. We can see that there is no global maximum either, since continuing to increase $ \ c \ $ still produces intersections between the surface and the plane.