Linked Questions

89
votes
20answers
22k views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
2
votes
3answers
8k views

Maximize volume of box in ellipsoid

I need to find the dimensions of the box with maximum volume (with faces parallel to the coordinate planes) that can be inscribed in ellipsoid $$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$$ ...
6
votes
5answers
1k views

How to show that any rectangle in ellipse must be oriented parallel to axes?

A problem which is often given as an exercise for students learning about calculus and finding extrema, is to find maximal possible area of a rectangle inside an ellipse. Such question was asked, for ...
1
vote
2answers
3k views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ $g(x,y,z)=x^2+2y^...
2
votes
1answer
474 views

Optimization of parallelepiped inside an ellipsoid

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
2
votes
1answer
297 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
2
votes
1answer
347 views

maximum volume of a box inside an ellipsoid

What is the maximum volume of a box that can be placed inside an ellipsoid $\frac{x^2}{16}+\frac{y^2}{9}+\frac{z^2}{25}=1$ The volume of a box is $V=xyz$ so I need to find $x,y,z$ with respect to the ...