Expert Help with High School Geometry

I'm having trouble maximizing the norm of $||Ax||$ such that $Bx=b$. I set up a lagrange multiplier so that $L(x,\mathrm{\lambda <!--\; \lambda \; -->})=x^T(A^{T}A)x-<!--\; -\; -->(Bx-<!--\; -\; -->b{)}^T\mathrm{\lambda <!--\; \lambda \; -->}$. (Say that $B$ is not invertible but $b$ is in the colspace so that the problem isn't trivial)
${\mathrm{\nabla <!--\; \nabla \; -->}}_{x}L(x,\mathrm{\lambda <!--\; \lambda \; -->})=0\Rightarrow <!--\; \Rightarrow \; -->2(A^{T}A)x=B^T\mathrm{\lambda <!--\; \lambda \; -->}$
and ${\mathrm{\nabla <!--\; \nabla \; -->}}_{\mathrm{\lambda <!--\; \lambda \; -->}}L(x,\mathrm{\lambda <!--\; \lambda \; -->})=0\Rightarrow <!--\; \Rightarrow \; -->Bx=b$.
If I assume that $A$ has full rank, then $x=\frac{1}{2}(A^{T}A{)}^{-<!--\; -\; -->1}{B}^T\mathrm{\lambda <!--\; \lambda \; -->}$.
If I plug it into the constraint, then $B(A^{T}A{)}^{-<!--\; -\; -->1}{B}^T\mathrm{\lambda <!--\; \lambda \; -->}=2b$. How do I proceed solving for $\mathrm{\lambda <!--\; \lambda \; -->}$ here?

Algebra Logical Pythagorean theorem help
A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. Find the length of the wire and the height of the pole.
Hint: Use the pythagorean theorem then set up a quadratic equation equal to zero and solve. I got c=17 and c=5 but I don't know what the length of the wire is or the height of the pole.

I was just wondering what the steps one would take to maximize the utility of a function of the form U(X,Y) = min{X,Y} + X subject to income I = $p_x$X + $p_y$Y where $p_x$ is the price of X and $p_y$ is the price of Y. I understand that normally this would be solved using the Lagrangian method but obviously you cannot differentiate this particular function.

Consider the following maximization problem.
$\begin{array}{rl}& \underset{x\in <!--\; \in \; -->{\mathbb{R}}^n}{max}{x}^{T}Px\\ & a_i\le <!--\; \le \; -->x_i\le <!--\; \le \; -->b_i,\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->{\mathbb{Z}}_{[1,n]},\end{array}$
where $x_i$ is the $i$-th scalar component of vector $x$ and $P>0$ is positive definite. As the objective is a convex function, the maximizer should be where all constraints are active, i.e. at an extreme point of the domain of the form $x_i^{\star <!--\; \star \; -->}=a_i$ or $x_i^{\star <!--\; \star \; -->}=b_i$. Is the global maximizer the one of such points such that its norm ${x^{\star <!--\; \star \; -->}}^T{x}^{\star <!--\; \star \; -->}$ is maximal?

Vector addition and Pythagorean theorem
Finding length or magnitude using vector addition and the Pythagorean theorem.
I am trying to understand why vector addition and the Pythagorean theorem are giving different results?

Vector Addition :
According to diagram (A) : $\stackrel{\to <!--\; \to \; -->}{a}+\stackrel{\to <!--\; \to \; -->}{b}=\stackrel{\to <!--\; \to \; -->}{c}$now suppose : magnitude of $\stackrel{\to <!--\; \to \; -->}{a}=3$, magnitude of $\stackrel{\to <!--\; \to \; -->}{b}=4$ then
$\Vert <!--\; \Vert \; -->\stackrel{\to <!--\; \to \; -->}{c}\Vert <!--\; \Vert \; -->=\Vert <!--\; \Vert \; -->\stackrel{\to <!--\; \to \; -->}{a}\Vert <!--\; \Vert \; -->+\Vert <!--\; \Vert \; -->\stackrel{\to <!--\; \to \; -->}{b}\Vert <!--\; \Vert \; -->$
$\Vert <!--\; \Vert \; -->\stackrel{\to <!--\; \to \; -->}{c}\Vert <!--\; \Vert \; -->=3+4$
magnitude of c = 7
Pythagorean Theorem
Now when we consider this as a triangle shown in diagram (B)
Similarly, length of a=3, length of b=4
so according to Pythagorean theorem
${\text{hypotenuse}}^2={\text{opposite}}^2+{\text{adjacent}}^2$
$c^2=a^2+b^2$i.e.
$c^2=3^2+4^2$
$c=\sqrt{9+16}=\sqrt{25}$
length of c = 5
Why there is inconsistency in the results? I am doing something wrong?
Thanks!

What is the easiest way to see that the path
$\underset{\_<!--\; \_\; -->}{r}:\mathbb{R}\to <!--\; \to \; -->{\mathbb{R}}^3:t\mapsto <!--\; \mapsto \; -->(\sin t,\cos t,\cos t)$
traces out an ellipse in the plane $y=z$ ?

I am reading a paper and they mention a hump function maximization: I am trying to prove the point of maximization:
$m=(1-<!--\; -\; -->x{)}^{1-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}}$ where $x,\mathrm{\sigma <!--\; \sigma \; -->}\in <!--\; \in \; -->[0,1]$
It is said that m is a hump-shaped function of x maximized at $x=\mathrm{\sigma <!--\; \sigma \; -->}$, where $\mathrm{\sigma <!--\; \sigma \; -->}$ is a parameter that is assumed to be fixed here.

My attempt:
First derivative with respect to x: $(1-<!--\; -\; -->x{)}^{-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}}+(1-<!--\; -\; -->x{)}^{1-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}-<!--\; -\; -->1}=0$
$1+(1-<!--\; -\; -->x{)}^1{x}^{-<!--\; -\; -->1}=0$
Second derivative with respect to x: $-<!--\; -\; -->(1-<!--\; -\; -->x{)}^{-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}-<!--\; -\; -->1}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}}+(1-<!--\; -\; -->x{)}^{-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}-<!--\; -\; -->1}-<!--\; -\; -->(1-<!--\; -\; -->x{)}^{-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}-<!--\; -\; -->1}+(1-<!--\; -\; -->x{)}^{1-<!--\; -\; -->\mathrm{\sigma <!--\; \sigma \; -->}}{x}^{\mathrm{\sigma <!--\; \sigma \; -->}-<!--\; -\; -->2}$
I couldn't get to the given result based on the above. Any clarification would be appreciated!

I am trying to find the maximum of a hermitian positive definite quadratic form $xQ{x}^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\le <!--\; \le \; -->1$, $i=1,\dots <!--\; \dots \; -->,n$ where $x=(x_1,x_2,\dots <!--\; \dots \; -->,x_n)\in <!--\; \in \; -->{\mathbb{C}}^n$.

This is the problem of minimizing a concave function over a convex domain. I have read that this problem is NP-hard but there exist some bounds on the optimum. What global optimization technique would your recommend to tackle this problem numerically? Since I am new to the field of optimization, I would appreciate every answer, thanks!

How does the Pythagorean theorem describe a circle?
The Pythagorean theorem states, for a right triangle with legs a,b and hypotenuse c,
$a^2+b^2=c^2$
By replacing c with r, radius this equation becomes the equation of circle at centre (0,0).
How does Pythagoras' equation end up describing the circle?

I have an exam in a few hours. I need to understand the solution to the following question

Find the Maximal to the the following $2x_1+3x_2$ is the objective function.

The constraints are
$4x_1+3x_2\le 600;$
$x_1+x_2\le 160;$
$3x_1+7x_2\le 840;$
$x_1,x_2\ge 0.$
Also the answer should be in the "Dual". If its possible please do it in the Algebraic method. If not I would just like the solution using the tableau method and how do you arrive to the solution.(PS: Any help would be great. )

You are a jeweler who sells necklaces and rings. Each necklace takes 4 ounces of gold and 2 diamonds to produce, each ring takes 1 ounce of gold and 3 diamonds to produce. You have 80 ounces of gold and 90 diamonds. You make a profit of 60 dollars per necklace you sell and 30 dollars per ring you sell, and want to figure out how many necklaces and rings to produce to maximize your profits.


Clearly this is a maximization problem, and this is how I formulated it.

Let $n$, $r$, $g$, and $d$ represent units of necklaces, rings, gold, and diamonds, respectively.

Then, $n=4g+2d$, and $r=g+3d$.

Our profit can be defined by $p=60n+30r$.

Our constraints are $g\le <!--\; \le \; -->80$, and $d\le <!--\; \le \; -->90$.

I tried to approach solving this by substituting n and r for their production functions in the profit function. But, on second thought, I'm not sure if plugging them in maintains an equivalent profit function, because the gold/diamond amounts are tied together for each necklace/ring made.

So, I'm not sure if $p=60n+30r$ and $p=60(4g+2d)+30(g+3d)$ are equivalent statements.

Am I approaching this problem correctly? If not, how should I think about the problem? Thank you.

We are given a ratio:
$\frac{g(x)}{f(x)}$
where:
$g(x)\in <!--\; \in \; -->{\mathbb{R}}^{+}$
$f(x)\in <!--\; \in \; -->\mathbb{N}\cap <!--\; \cap \; -->f(x)\ge <!--\; \ge \; -->2$
So $g(x)$ returns values in $[0,+\mathrm{\infty <!--\; \infty \; -->}]$ while $f(x)$ returns values in $\{2,3,4,\dots <!--\; \dots \; -->\}$.

I am looking for a confirmation about a very simple question: if I maximize ${\displaystyle \frac{g(x)}{f(x)}}$, do I also maximize ${\displaystyle \frac{g(x)}{f(x)-<!--\; -\; -->1}}$ in this very particular case?

Find the equation of angle bisector passing through quadrant containing (2,3).
$L_1\Rightarrow 2x+y-<!--\; -\; -->1=0$
$L_2\Rightarrow 2x-<!--\; -\; -->y+3=0$

I need to learn how to find the properties (foci, vertex, etc) of a conic given its equation, I know how to do it when the coeficient of $xy=0$ , but I have troubles in the other case. Can someone recommend me a book, website o document where I can learn how to find the properties of a conic section given its equation please?

We need to prove that when we apply the Newton-Raphson method to strictly quadratic concave function. It will converge in one step.

How to apply this method to maximization of
$f(x)=4\cdot <!--\; \cdot \; -->x_1+6\cdot <!--\; \cdot \; -->x_2-<!--\; -\; -->2\cdot <!--\; \cdot \; -->x_1^2-<!--\; -\; -->2\cdot <!--\; \cdot \; -->x_1\cdot <!--\; \cdot \; -->x_2-<!--\; -\; -->2\cdot <!--\; \cdot \; -->x_2^2$
I did not understand how to apply method to this function? what should be the interval?

I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywords to find it. Please help.

Say there is a function $G(k,x)={\int <!--\; \int \; -->}_a^{x}f(k,t)dt$, and I wish to maximize $G(k,x)$ w.r.t. $k$. Under what conditions is this maximization problem equivalent to maximizing $f(k,t)$ w.r.t. $k$?

In short, when is the following true:
$\underset{k}{max}\{{\int <!--\; \int \; -->}_a^{x}f(k,t)\}dt\iff <!--\; \iff \; -->\underset{k}{max}\{f(k,t)\}$

A coffee company sells two types of breakfast blends. They have on hand 132 kg of dark roast and 84 kg of hazelnut. One breakfast blend will contain half dark roast and half hazelnut and will sell for $\mathrm{\$<!--\; \$\; -->}7$ per kg. The other breakfast blend will contain 3/4 dark roast and 1/4 hazelnut and will sell for $\mathrm{\$<!--\; \$\; -->}9.50$ per kg.

My approach:
I see this is an optimization problem and need to show that $132x_1+84x_2$ needs to be optimized but confused on the other constraints
I used the convex optimization polygon theory of checking on the endpoints and computing the maximal value, can someone check and elaborate further on the reply below?

I would like to calculate the arc length of a circle segment, i.e. I know the start coordinates $(x/y)$ of the circle segment, the end coordinates $(x/y)$ and the x and y distances from the starting point to the center point of the circle segment.
I know that I can calculate the circumference with 2 * radius * PI. Consequently, I would have to calculate the radius and the angle of the circle segment via Pythagorean theorem and sin and cos. My question: Is there a simple formula where I just have to put in start-coordinates, end-coordinates and the circle origin point coordinates?
Thanks.

Find equation of circle passing through intersection of circle
$S=x^2+y^2-<!--\; -\; -->12x-<!--\; -\; -->4y-<!--\; -\; -->10=0$
and
$L:3x+y=10$ and having radius equal to that of circle S.