Discover Maximization Techniques in Geometry Triangles

Erroneously Finding the Lagrange Error Bound
Consider $f(x)=\sin <!--\; \; -->(5x+{\displaystyle \frac{\mathrm{\pi <!--\; \pi \; -->}}{4}})$ and let P(x) be the third-degree Taylor polynomial for f about 0. I
am asked to find the Lagrange error bound to show that |(f(1/10)−P(1/10))|<1/100. Because P(x) is a third-degree polynomial, I know the difference is in the fourth degree term.

M maximal iff $\stackrel{¯<!--\; ¯\; -->}{M}$ is maximal
We have a ring R and I an ideal of R. Let $M$ be an ideal of R containing I. Let $\stackrel{¯<!--\; ¯\; -->}{M}$ be M/I and $\stackrel{¯<!--\; ¯\; -->}{R}$ be R/I. Prove that $M$ is maximal if and only if $\stackrel{¯<!--\; ¯\; -->}{M}$ is maximal.
I think I get the general idea, that $M$ not maximal means there exists a bigger $M$′ containing $M$ and so $M^{\prime }/I$contains $M/I$. How do I formalize this idea? And how do I show the other direction?

The Euclidean algorithm starts with two numbers m and n, then computes
$$\begin{gathered} m=n(q)+r_1 \\ n=(r_1)(q_2)+r_2 \\ {r}_1=(r_2)(q_3)+r_3 \\ {r}_2=(r_3)(q_4)+r_4 \end{gathered}$$
and so on, until
$$\begin{gathered} r_{n\mathrm{\_<!--\; \_\; -->}1}=(r_n)(q_{n+1})+r_{n+1} \\ {r}_n=(r_{n+1})(q_{n+2}) \end{gathered}$$
and $r_{n+1}$ is the desiblack gcd. Prove that r_n+1 is at least a divisor of both m and n.

Consider the primes arranged in the usual order 2, 3, 5, 7... It is conjectublack that beginning with 3, every other prime can be composed of the addition and subtraction of all smaller primes (and 1), each taken once. For example:
3 = 1 + 2
7 = 1 - 2 + 3 + 5
13 = 1 + 2 - 3 - 5 + 7 + 11 = -1 + 2 + 3 + 5 - 7 + 11
A. Show that this also holds for 19.
B. Show that this doesn't hold for the first "inbetween" prime 5

Is there any relationship similar to the following. Let $X$ be the maximum of functions $f_1(x)+f_2(x)$. Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$. Is there any relationship between $X$ and $X_1$ and $X_2$?

For example can we say under what condition will $X$ be in-between $X_1$ and $X_2$, $X_1\le <!--\; \le \; -->X\le <!--\; \le \; -->X_2$.

Any reference would be greatly appreciated.

I would like to maximize the function:
$\frac{1}{2}\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}|<!--\; |\; -->x_i-<!--\; -\; -->\frac{1}{N}|<!--\; |\; -->$
under the constrains $\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}{x}_i=1$ and $\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->(1,...,N)$
I have done some test for small values of $N$ and I have the feeling that the solution is $1-<!--\; -\; -->\frac{1}{N}$ but I can't figure out how to solve it analytically.

Find the maximum profit corresponding to a demand function of $p=36-<!--\; -\; -->4x$ and a total $\text{cost}\text{function}=2x^2+6$

Hello. Can you kindly help me solve this problem? Thank you in advance

Find the function $s(x)$ such that $s(x)$ maximizes
${\int <!--\; \int \; -->}_0^{s^{-<!--\; -\; -->1}(k)}s(x)dx$
where $x\in <!--\; \in \; -->[0,10]$, $s(x)\in <!--\; \in \; -->[0,1]$, and $k\in <!--\; \in \; -->[0,1]$ ($k$ is a constant).

I am an economist with some math background but not strong enough to solve this. I'm trying to solve:
$\begin{array}{r}\underset{x}{max}f(x,y(x))=a\cdot <!--\; \cdot \; -->y(x)+b\cdot <!--\; \cdot \; -->g(x)\end{array}$
where
$\begin{array}{r}y(x)=1(h(x)>0).\end{array}$
$h(x)$ is a linear function of $x$ and $a,b\in <!--\; \in \; -->\mathbb{R}$ are constants. $x\in <!--\; \in \; -->X$ and $X$ is a compact and continuous subset of $[0,1]$. $g(x)$ is a concave and differentiable function.

Now, my problem is that the optimal choice of $x$ depends on the value of $y$($x$), which depends on $x$.

I would like to know how to solve for, if possible, a closed-form solution. If it is impossible, what kind of algorithm should I use, and where can I find the essential readings to learn them.

EDIT: Thanks Robert for his great answer. Now I would just like to ask this follow-up question:

Can this method of solving for the maximum be adopted in a dynamic programming version of this problem? Say I want to maximise$V(x_t)=\underset{x_t}{max}f(x_t,y_t(x_t))+\mathrm{\delta <!--\; \delta \; -->}V(x_{t+1})$ but the constant $a$ now becomes a function of $x_{t-1}$? $\mathrm{\delta <!--\; \delta \; -->}$ is a discount factor.

I was trying to solve the problem A maximization problem when I ask myself if the general problem
$\begin{array}{c}maximizef(\mathbf{X}{)}^p+g(\mathbf{X}{)}^p\\ s.t.\mathbf{X}\in <!--\; \in \; -->K\subseteq <!--\; \subseteq \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->n},\end{array}$
is equivalent to
$\begin{array}{c}maximizef(\mathbf{X})+g(\mathbf{X})\\ s.t.\mathbf{X}\in <!--\; \in \; -->K\subseteq <!--\; \subseteq \; -->{\mathbb{R}}^{m\times <!--\; \times \; -->n},\end{array}$
when the scalar functions $f(\mathbf{X})$ and $g(\mathbf{X})$ are nonnegative on $K$, and $p>0$.
Is this true? If not, how to find a counterexample?

I`m trying to solve a maximization problem:
$\underset{\mathbf{p}}{max}w\text{ln}(\underset{i=1}{\overset{I}{\sum <!--\; \sum \; -->}}{p}_i{a}_i)-<!--\; -\; -->\underset{i=1}{\overset{I}{\sum <!--\; \sum \; -->}}{p}_i{d}_i,$
where $p_i\in <!--\; \in \; -->\{0,1\}$ is binary variable and $\underset{i=1}{\overset{I}{\sum <!--\; \sum \; -->}}{p}_i=1$. I need to find optimal $p_i^{\ast <!--\; \ast \; -->}$, where
${\mathbf{p}}^{\ast <!--\; \ast \; -->}=\arg <!--\; \; -->\underset{\mathbf{p}}{max}w\text{ln}(\underset{i=1}{\overset{I}{\sum <!--\; \sum \; -->}}{p}_i{a}_i)-<!--\; -\; -->\underset{i=1}{\overset{I}{\sum <!--\; \sum \; -->}}{p}_i{d}_i$
Is there any way to obtain the closed-form of $p_i^{\ast <!--\; \ast \; -->}$ as the function of $w$, $a_i$, and $d_i$?

I have to solve this optimization problem
$\underset{x}{max}(AB-<!--\; -\; -->\frac{xC}{D}(E+F))$
subject to
$\frac{AB}{x}-<!--\; -\; -->\frac{C}{D}(E+F)\le <!--\; \le \; -->G$
and
$0<x\le <!--\; \le \; -->\frac{ABD}{C(E+F)}.$
How can I solve it?

Let $\{X_1,\dots <!--\; \dots \; -->,X_K\}$ is a set of random matrices, where $X_k\in <!--\; \in \; -->{\mathbb{R}}^{M\times <!--\; \times \; -->N},k=1,\dots <!--\; \dots \; -->,K$, and $U\in <!--\; \in \; -->{\mathbb{R}}^{M\times <!--\; \times \; -->r}$ and $V\in <!--\; \in \; -->{\mathbb{R}}^{N\times <!--\; \times \; -->r}$ are two matrices containing orthogonal columns (i.e., $U^{\mathrm{\top <!--\; \top \; -->}}U=I,V^{\mathrm{\top <!--\; \top \; -->}}V=I$). I was wondering, if the following question has a analytical solution:
${\displaystyle \underset{U,V}{max}\underset{k=1}{\overset{K}{\sum <!--\; \sum \; -->}}\Vert <!--\; \Vert \; -->U^{\mathrm{\top <!--\; \top \; -->}}{X}_{k}V{\Vert <!--\; \Vert \; -->}_F^2}$
If not, how should I solve it? Alternating optimization?

(At first, I thought it may be related to the SVD of the sum of the matrices $\{X_k\}$, but so far I have no hint to prove it.)

for three variables,
$$\begin{gathered} maxf(x,y,z)=xyz \\ \text{s.t.}(\frac{x}{a}{)}^2+(\frac{y}{b}{)}^2+(\frac{z}{c}{)}^2=1 \end{gathered}$$
where $a,b,c$ are constant
how to solve the maximization optimization problem?
thank you for helpin

I have the following maximization problem
$\underset{h}{max}{m}_1+10(h{)}^{1/4}+h+m_2-<!--\; -\; -->2(h{)}^{1/4}+m_3-<!--\; -\; -->(h{)}^{1/4}$
where $m_1,m_2,m_3$ are three fixed values. The FOC for a maximum is
${\displaystyle \frac{10}{4}}{h}^{-<!--\; -\; -->3/4}+1-<!--\; -\; -->{\displaystyle \frac{1}{2}}{h}^{-<!--\; -\; -->3/4}-<!--\; -\; -->{\displaystyle \frac{1}{4}}{h}^{-<!--\; -\; -->3/4}=0$
Rearranging
${\displaystyle \frac{7}{4}}\cdot <!--\; \cdot \; -->h^{-<!--\; -\; -->3/4}=-<!--\; -\; -->1$
Well, now I have no idea... how can I go on? How can I say which level of $h$ maximizes my problem? Of course, I don't want you to solve, but I just would like to get a hint! Thank you in advance!

Estimating the parameters of gaussians to fit a lot of samples can be do with Exceptation Maximization, for instance if we want to fit two gaussian on points, to have the clusters $a$ and $b$. (1)
$b_i=P(b|x_i)=\frac{P(x_i|b)P(b)}{P(x_i|b)P(b)+P(x_i|b)P(a)}$
Here $P(b)$ is the prior that depicts the overall importance of the $b$ cluster.
This prior is then updated for the next step, according on how many the points belongs to the $b$ cluster:
$P(b)=\frac{1}{n}\underset{i}{\sum <!--\; \sum \; -->}{b}_i$
However, what is the value of the prior $P(b)$ on the first iteration of the algorithm?

Maximize
$f(x,y,z)=xy+z^2,$
while $2x-<!--\; -\; -->y=0$ and $x+z=0$. Lagrange doesnt seem to work.

Consider the problem for some vectors $v,m\in <!--\; \in \; -->{\mathbb{R}}^n:$

$f(v)=(v^{T}m{)}^2$

w.r.t $\Vert <!--\; \Vert \; -->v{\Vert <!--\; \Vert \; -->}^2=1$

I want to maximize f

If I consider the lagrangian, I get:

$L(v)=(v^{T}m{)}^2+\mathrm{\lambda <!--\; \lambda \; -->}(1-<!--\; -\; -->\Vert <!--\; \Vert \; -->v{\Vert <!--\; \Vert \; -->}^2)$

Taking derivative, I get: $2m{m}^{T}v-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}2v=0$ Therefore $m{m}^{T}v=\mathrm{\lambda <!--\; \lambda \; -->}v(\ast <!--\; \ast \; -->)$

Multiplying by $v^T$ from left, I end up with

$(v^{T}m{)}^2=\mathrm{\lambda <!--\; \lambda \; -->}$

If I put that in(*), I do cannot simplfy that.

Is there a trick I can apply?

Let $(c_1,c_2)$ be a fixed point in ${\mathbb{R}}^2$. How to maximize $|x_1{x}_2-<!--\; -\; -->c_1{c}_2|$ subject to the condition that $(x_1-<!--\; -\; -->c_1{)}^2+(x_2-<!--\; -\; -->c_2{)}^2<1$. i.e.
$\underset{(x_1,x_2)\in <!--\; \in \; -->{\mathbb{R}}^2:(x_1-<!--\; -\; -->c_1{)}^2+(x_2-<!--\; -\; -->c_2{)}^2<1}{sup}|x_1{x}_2-<!--\; -\; -->c_1{c}_2|=?$
Note : This is a generalization of the problem of maximizing $|x_1{x}_2|$ subject to the condition $x_1^2+x_2^2<1$. I'm stuck with it. Any help would be much appreciated.

I'm facing this problem:
$\underset{x\in <!--\; \in \; -->{\mathbb{R}}_{+}^3}{min}max\{\frac{\underset{i=1}{\overset{3}{\sum <!--\; \sum \; -->}}{x}_i^2-<!--\; -\; -->2x_1{x}_3}{{\left(\underset{i=1}{\overset{3}{\sum <!--\; \sum \; -->}}{x}_i\right)}^2},\frac{\underset{i=1}{\overset{3}{\sum <!--\; \sum \; -->}}{x}_i^2+2(x_1{x}_3-<!--\; -\; -->x_1{x}_2+x_2{x}_3)}{{\left(\underset{i=1}{\overset{3}{\sum <!--\; \sum \; -->}}{x}_i\right)}^2}\}$
I don't know how to deal with inner max and choose one of two!

I'm trying to use $max(A,B)\ge <!--\; \ge \; -->\frac{1}{2}(A+B)$! Do you have any idea?