What is the minimum volume bounded by the planes $x=0, y=0, z=0$ and a plane which is tangent to the ellipsoid
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1$$ where $x,y,z>0$
I only know that when we want to find maximum and minimum, there is constraint $g$ such that $\nabla f = a\nabla g$ where $a$ is Lagrange multiplier.
What is the constraint and what is $f$?
On
Here is an illustration of the problem at hand.
You want to find the the volume of the tetrahedron formed by the $x=y=z=0$ planes, and the plane tangent to the ellipsoid at some point on its surface, $(x,y,z)$. This volume, of course, will vary with different points on the surface of the ellipse, so the constraint function must be the following equality
$$\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2=1$$
Now the vector normal to the surface of the ellipsoid at some point $(x,y,z)$ is given by the gradient of its equation.
$$\nabla g (x,y,z) = \left<\frac{2x}{a^2},\frac{2y}{b^2},\frac{2z}{c^2}\right>$$
The equation for the plane is tangent to the ellipsoid at this point is easily worked out to be
$$\frac{x}{a^2}x'+\frac{y}{b^2}b'+\frac{z}{c^2}z'=1$$
Now, finally, for the volume that this plane bounds within the first quadrant, we make use of the fact that this volume is simply $1/6$ the volume of the rectangular prism whose side lengths are given by the points of where the plane intersects the three axes.
$$V=\frac{(abc)^2}{6xyz}$$
And that is your function $f$ which you are trying to minimize.
$$f=V$$
$$g\rightarrow \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2=1$$
The constraint is the tangency point, which must be on the ellipsoid, and the function to minimise is the volume created. The resulting "tangent plane" can be defined in various ways, as long as when the "tangency point" is on the ellipsoid, it must equal to the real tangent plane to that point. For example, if the ellipsoid is the unit sphere, $(p,q,r)\cdot(x,y,z)=1$ is one possible way, where $(p,q,r)$ is the point that varies. I think that by scaling the ellipsoid, you can actually reduce the problem to the unit sphere case.