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I need to prove the maxima of the following summation, using Lagrange.
$\underset{x_m}{max}(\sum _m{a}_{m}log(x_m))$
s.t.
$0\le x_m\le 1$
$\sum _m{x}_m=1$
The solution is a closed form $x_m=\frac{a_m}{\sum _m{a}_m}$ .

I formulated the Lagrange equation but I am confused about the signs and the multipliers.

$L(x,\lambda ,\mu )=\sum _m{a}_{m}log(x_m)+\sum _m{\lambda }_m(1-x_m)+\mu (\sum _m{x}_m-1)$, Is this formulation correct ? what is wrong ?

note: only one $\mu$ for one constraint.

Let $X=(X_1,...,X_n)$ be a vector of $n$ random variables. Consider the following maximization problem:
$\underset{a,b}{max}\mathrm{C}\mathrm{o}\mathrm{v}(a\cdot X,b\cdot X)$ under the constraint that $\Vert a{\Vert }_2=\Vert b{\Vert }_2=1$.
($a\cdot X$ is the dot product between $a$ and $X$). Would it be true that there is a solution to this maximization problem such that $a=b$?
Thanks.

This is a utility maximzation problem

maximize $x^a+y^b$ subject to $p_{1}x+p_{2}y=w$ (utility maximization problem)

Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see it. many thanks!!!

Maximize the value of the function
$z=\frac{ab+c}{a+b+c},$
where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you.

A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a sector with vertex angle theta and welding the straight edges of the straight edges of the remaining piece together. Find theta so that the resulting cone has the largest possible volume.

Specifically, the question is asked in the context of wanting derivatives, multiple max/min equations, and hopefully more calc rather than trig or geo.

I have gotten as far as using 10m as the hypotenuse for a triangle formed by the height of the cone, radius of the base of the cone, and slant. I'm not sure where to go from there, because I can't determine how to find height and/or radius, without which I'm not sure I can continue.

Show that $f(x_1^{\ast },...x_n^{\ast })=max\{f(x_1,...,x_n):(x_1,...,x_n)\in \mathrm{\Omega }\}$ if and only if $-f(x_1^{\ast },...x_n^{\ast })=min\{-f(x_1,...,x_n):(x_1,...,x_n)\in \mathrm{\Omega }\}$
I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of $f$. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts?

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as $y=b-ax$, and $y=c-d{x}^2$, but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.