Expert Help with High School Geometry

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

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

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

note: only one $\mathrm{\mu <!--\; \mu \; -->}$ for one constraint.

Converting between the $y=e+dx$ and $ax+by=c$ forms of a line equation

I have a non-linear maximization problem and I want to convert it to be a minimization problem, can I do so by multiplying it by a negative sign, or is that wrong; and if that is wrong what should I do?

Plane with normal vector
a) If unit normal vector is $(a_1,b_1,c_1),$ , then, how the point $P_1$ on the plane becomes $(D{a}_1,D{b}_1,D{c}_1)?$ ?
b) If unit normal is $(1/3,2/3,2/3)$ then $P_1$ becomes $(2/3,4/3,4/3)$ Where $D=2.$ We know that normal vector began on the plane at point $P_1$ and ends at $(1/3,2/3,2/3).$ My questions is how unit normal vector coordinates value less than $P_1$ coordinates value, because unit normal vector pointing outside of the plane it should be greater coordinates value than $P_1$ ?

We have a function:
$f(x,y,z)=x\cos <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}t+y+{\mathrm{\varphi <!--\; \varphi \; -->}}_1)+z\cos <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}t+y+{\mathrm{\varphi <!--\; \varphi \; -->}}_2).$
I want to solve the following maximization problems:
$\underset{x,y,z}{max}\underset{t}{max}f(x,y,z)\text{or}\underset{x,y,z}{max}{\int <!--\; \int \; -->}_{<T>}f(x,y,z)dt.$
Surely, $x$ and $z$ are nonnegative, and $y$ is in $[0,2\mathrm{\pi <!--\; \pi \; -->})$.

Can someone give me hints for solving the problem, or let me know some references to solve this types of problem?

Each cow weighs 300kg. The cost of maintaining a cow is 20€ per day. The rate of change of the cows weight is 4kg a day. The market price is currently 50€ per kg and it is falling 50 cents a day. How much time should the cattleman wait to sell the cows? How much extra money does he make by waiting?

Let $f(x,y):\mathbb{R}\times <!--\; \times \; -->\mathbb{R}\mapsto <!--\; \mapsto \; -->\mathbb{R}$ be a continuous and differentiable function. When can we claim that the following holds true:
$\frac{d}{dx}\underset{y\in <!--\; \in \; -->\mathbb{R}}{max}f(x,y)=\underset{y\in <!--\; \in \; -->\mathbb{R}}{max}\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}x}f(x,y)$
assuming that both $\underset{y\in <!--\; \in \; -->\mathbb{R}}{max}f(x,y)$ and $\underset{y\in <!--\; \in \; -->\mathbb{R}}{max}\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}x}f(x,y)$ exists.

I have read a paper where, at some point, the following optimization problem appears:
$\begin{array}{rl}& \underset{\mathit{x}}{max}{\mathit{w}}^T\mathit{x}\\ & \text{subject to }{\Vert \mathit{x}\Vert }_{\mathrm{\infty <!--\; \infty \; -->}}\le <!--\; \le \; -->\mathrm{ϵ<!--\; ϵ\; -->}\end{array}$
where $\mathit{x}$ and $\mathit{w}$ are real vectors and $\mathrm{ϵ<!--\; ϵ\; -->}$ is some given positive constant. The authors claim that the solution for this problem is $\mathit{x}=\text{sign}(\mathit{w})$, which seems clearly wrong to me, since then we would have ${\Vert \mathit{x}\Vert }_{\mathrm{\infty <!--\; \infty \; -->}}=1$.

I believe the correct answer is $\mathit{x}=\mathrm{ϵ<!--\; ϵ\; -->}\text{sign}(\mathit{w})$. However, I am not being able to prove it. I have tried KKT multipliers, with no success. Any ideas?

Consider that I have two items. Let $P_i(x_i)$ denote the profit I get from item i by taking $x_i$ units of it (assume that $x_i$ can be a real number). Let $W_i(x_i)$ denote the weight consumed by item i when I take $x_i$ units of this. Assume that both profit and weight functions are monotonically increasing in the quantity of items picked. The problem I am interested in is
$\begin{array}{rl}\underset{x_1,x_2}{max}& P_1(x_1)+P_2(x_2)\\ s.t.& W_1(x_1)+W_2(x_2)\le <!--\; \le \; -->W\end{array}$
where $W$ is total weight allowed. Thus I need to find $x_i$ (quantity of item $i$) such that the total profit is maximized while keeping the total weight under a given quantity. Let $x_1^{\ast <!--\; \ast \; -->}$ and $x_2^{\ast <!--\; \ast \; -->}$ be the solution to this problem. Is it true that
$\frac{P_1(x_1^{\ast <!--\; \ast \; -->})}{W_1(x_1^{\ast <!--\; \ast \; -->})}=\frac{P_2(x_2^{\ast <!--\; \ast \; -->})}{W_2(x_2^{\ast <!--\; \ast \; -->})}$
Note that ratio is essentially "Profit Per unit weight of an item". Assume that this were different at the optimum and the first item had the larger ratio. Thus I can get more profit per unit weight out of the first item. Then increasing quantity of that should increase profit. In order to balance the weights, I decrease the second items quantity decreasing its profit as well. But since the increase in profits from item 1 is higher, this should compensate for loss from item 2 and make the profits even more higher. What is flawed with this logic?

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 <!--\; \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 ${\displaystyle \frac{\mathrm{\partial <!--\; \partial \; -->}u(x_i)}{\mathrm{\partial <!--\; \partial \; -->}{x}_i}}$.

For some strictly convex functions $f_1(x_1),f_2(x_2),...,f_n(x_n)$, it seems intuitive that for a given sum of $x_1+x_2+...+x_n=X$, where $x_1$, $x_2$,..., $x_n\ge <!--\; \ge \; -->0$
$\mathrm{\exists <!--\; \exists \; -->}k\in <!--\; \in \; -->\{1,2,\dots <!--\; \dots \; -->,n\},f_k(X)=max\{\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{f}_i(x_i):\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i=X\text{ and }\mathrm{\forall <!--\; \forall \; -->}i,x_i\ge <!--\; \ge \; -->0\}$
Since these functions are convex, the sum of these functions is greatest when the sum of arguments is equal to the argument of just one function. This seems intuitive, but I can't think of how to argue this in formal maths. Any suggestions?

How can I find ${\mathrm{\lambda <!--\; \lambda \; -->}}_H$ and ${\mathrm{\lambda <!--\; \lambda \; -->}}_T$ such that
$\underset{0\le <!--\; \le \; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_H,{\mathrm{\lambda <!--\; \lambda \; -->}}_T\le <!--\; \le \; -->1}{max}\{\frac{4.6575342\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}}{2.1722965\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}+{\mathrm{\lambda <!--\; \lambda \; -->}}_H},\frac{1.0958904\times <!--\; \times \; -->{10}^{-<!--\; -\; -->2}}{3.4311896\times <!--\; \times \; -->{10}^{-<!--\; -\; -->4}+{\mathrm{\lambda <!--\; \lambda \; -->}}_T}\}<1?$
Is this problem equivalent to finding $x$ and $y$ such that
$\underset{0\le <!--\; \le \; -->x,y\le <!--\; \le \; -->1}{min}\{.4664048+(2147.0588450)x,0.0313096+(91.2500009)y\}\ge <!--\; \ge \; -->1?$

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 $f(x_1^{\ast <!--\; \ast \; -->},...x_n^{\ast <!--\; \ast \; -->})=max\{f(x_1,...,x_n):(x_1,...,x_n)\in <!--\; \in \; -->\mathrm{\Omega <!--\; \Omega \; -->}\}$ if and only if $-<!--\; -\; -->f(x_1^{\ast <!--\; \ast \; -->},...x_n^{\ast <!--\; \ast \; -->})=min\{-<!--\; -\; -->f(x_1,...,x_n):(x_1,...,x_n)\in <!--\; \in \; -->\mathrm{\Omega <!--\; \Omega \; -->}\}$
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?