Show that f ( <msubsup> x 1 &#x2217;<!-- ∗ --> </msubsup> , . .

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.

I try to maximize a quadratic function on box constraints as shown below
$\begin{array}{l}max{x}^{T}Qx\\ s.t.a\le x_i\le b\end{array}$
The length of vector $x$ is approximately 2000 and $Q$ is symmetric. I am wondering if there exists a solver (like Cplex or Gurobi) which can solve this maximization problem efficiently.

Statement:

A manufacturer has been selling 1000 television sets a week at $\mathrm{\$<!--\; \$\; -->}480$ each. A market survey indicates that for each $\mathrm{\$<!--\; \$\; -->}11$ rebate offered to a buyer, the number of sets sold will increase by 110 per week.

Questions :

a) Find the function representing the revenue R(x), where x is the number of $\mathrm{\$<!--\; \$\; -->}11$ rebates offered.

For this, I got $(110x+1000)(480-11x)$. Which is marked correct

b) How large rebate should the company offer to a buyer, in order to maximize its revenue?

For this I got 17.27. Which was incorrect. I then tried 840999, the sum total revenue at optimized levels

c) If it costs the manufacturer $\mathrm{\$<!--\; \$\; -->}160$ for each television set sold and there is a fixed cost of $\mathrm{\$<!--\; \$\; -->}80000$, how should the manufacturer set the size of the rebate to maximize its profit?

For this, I received an answer of 10, which was incorrect.

The maximization problem is:
Maximize $u(x_1,x_2)=min[a_1{x}_1,a_2{x}_2]\text{s.t.}{p}_1{x}_1+p_2{x}_2\le$$w$, in which $x_i,p_i$ is the amount and price of good $i$, $w$ is the total budget available.
What I have been told to deal with this min[.,.] function is to solve it graphically. It's very easy to see on the graph that the maximization happens when $a_1{x}_1=a_2{x}_2$. But I wonder if there is a way to solve this algebraically? I'm stumped at the first step, which is to derive ${\displaystyle \frac{\mathrm{\partial }u(x_i)}{\mathrm{\partial }{x}_i}}$.

Which are the most famous problems having an objective of maximizing a nonlinear convex function (or minimizing a concave function)? As far as I know such an objective with respect to linear constraints is np-hard.

I am trying to solve the following question:
$\text{Maximize }f(x_1,x_2,\dots ,x_n)=2\sum _{i=1}^n{x}_i^{t}A{x}_i+\sum _{i=1}^{n-1}\sum _{j>i}^n(x_i^{t}A{x}_j+x_j^{t}A{x}_i)$
subject to
$x_i^t{x}_j=\{\begin{array}{cc}1& i=j\\ -\frac{1}{n}& i\ne j\end{array}$
where xi's are column vectors ($m\times 1$ matrices) with real entries and A is an $m\times m$ ($(n<m)$) real symmetric matrix.

From some source, I know the answer as
$f_{max}=\frac{n+1}{n}\sum _{i=1}^n{\lambda }_i^{\downarrow },$
${\lambda }_i^{\downarrow }$ being the eigenvalues of $A$ sorted in non-increasing order (counting multiplicity). But I am unable to prove it. I will appreciate any help (preferably with established matrix inequality, or Lagrange's multiplier method).

I'm a first year Finance grad student and we're learning utility maximization problem now. We have been assuming that a solution to these problems exists. Are there any general theorem about the existence of maximization solution on ${\mathbb{R}}^n$? I have encountered once that "A utility maximization problem with a continuous utility function on a compact set has solution". Thanks in advance!