Expert Help with High School Geometry

Given a circle M of known center and radius and a right-angled triangle JKL with one vertex (J) at the circle's center and another vertex (K) at the circle's boundary, find the cartesian coordinates of the third vertex (L) (with respect to the center of the circle). The three sides of the triangle JKL are given, where one of the sides (A) is a radius of the circle. The coordinates for the vertex K are not known.

Let $\mathrm{△<!--\; △\; -->}ABC$ be a scalene triangle, I its incenter and G its centroid. Prove that the line GI intersects both (AB) and (AC) on the line segments (id est between the two points of extremities), if and only if there exists a $t\in <!--\; \in \; -->[0;1)$ for which we may write $a=tb+(1-<!--\; -\; -->t)c$ , where a, b, c are the sides of the triangle in trigonometric notation (id est $a=BC$ , $b=AC$ , $c=AB$ ).

Find the equation of an ellipse if its center is S(2, 1) and the edges of a triangle PQR are tangent lines to this ellipse. P(0, 0), Q(5, 0), R(0, 4).
My attempt: Let take a point on the line PQ. For example (m,0). Then we have an equation of a tangent line for this point: $(a_{11}m+a_1)x+(a_{12}m+a_2)y+(a_{1}m+a)=0$, where $a_{11}$ etc are coefficients of our ellipse: $a_{11}{x}^2+2a_{12}xy+a_{22}{y}^2+2a_{1}x+2a_{2}y+a=0$. Now if PQ: y = 0, then $(a_{11}m+a_1)=0$ , $a_{12}m+a_2=1$, $a_{1}m+a=0$ .I've tried this method for other 2 lines PR and RQ and I got 11 equations (including equations of a center)! Is there a better solution to this problem?

Edit: Removed solved in title, because I realize I need someone to check my work.
Ok, so the problem is a lot more straight forward than I originally approached it (which was a false statement -- so it was excluded).
Let R,S, x $\in <!--\; \in \; -->$ N with x $\le <!--\; \le \; -->$ R*S and 0 $0<<!--\; <\; -->$ R $\le <!--\; \le \; -->$ S. Next, define B as a multiplicative factor of x - c with c $\ge <!--\; \ge \; -->0$ 0 and B $\le <!--\; \le \; -->$ S such that $\frac{x-<!--\; -\; -->c}{B}=A\le <!--\; \le \; -->R$ and A $\in <!--\; \in \; -->$ N. What value of B maximizes A?

Without using Lagrange multipliers, find the maximum volume of a rectangular box inscribed in tetrahedron bounded by the coordinate planes and the plane 2x/5 + y + z = 1

I am self-teaching vector calculus but I got stuck on this question. Thanks for any help! :)

I have to maximize $U(x,y)=Min(ax+y,by+x)$ s.a $p_{1}x+p_{2}y=m$. I try the traditional solution for a leontieff $(a{x}_1+y=b{y}_1+x)$ function but I'm not sure.. beacause exist regions where one plan is under the other and only one of them is a minimun...

$$\begin{gathered} maxmin[\mathrm{\alpha <!--\; \alpha \; -->}{x}_1,\mathrm{\beta <!--\; \beta \; -->}{x}_2,\mathrm{\gamma <!--\; \gamma \; -->}{x}_3]\text{s.t.}{\mathrm{\lambda <!--\; \lambda \; -->}}_1{x}_1+{\mathrm{\lambda <!--\; \lambda \; -->}}_2{x}_2+{\mathrm{\lambda <!--\; \lambda \; -->}}_3{x}_3=c, \\ \mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->},\mathrm{\gamma <!--\; \gamma \; -->},{\mathrm{\lambda <!--\; \lambda \; -->}}_i,c\text{are constants} \end{gathered}$$
Well, that function is not differentiable , so what methods can be applied to solve for for the optimal values of $x_1$,$x_2$ and $x_3$? Is knowledge of the ${\mathrm{\lambda <!--\; \lambda \; -->}}^{\prime }s$ and $c$ necessary, to at least some degree, or does a general approach/solution exist?

I am given the following problem:
$A=\left[\begin{array}{ccc}4& 1& 1\\ 1& 2& 3\\ 1& 3& 2\end{array}\right]$
Find
$\underset{x}{max}\frac{|(Ax,x)|}{(x,x)}$
where $(.,.)$ is a dot product of vectors and the maximization is performed over all $x={\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^T\in <!--\; \in \; -->{\mathbb{R}}^3$, such that ${\displaystyle \underset{i=1}{\overset{3}{\sum <!--\; \sum \; -->}}{x}_i=0}$
I have found the eigenvectors for $A$ and they happen to match the sum criterion:
$E({\mathrm{\lambda <!--\; \lambda \; -->}}_1)=\text{span}({\left[\begin{array}{ccc}-<!--\; -\; -->2& 1& 1\end{array}\right]}^T)$
for ${\mathrm{\lambda <!--\; \lambda \; -->}}_1=3$ and
$E({\mathrm{\lambda <!--\; \lambda \; -->}}_2)=\text{span}({\left[\begin{array}{ccc}0& -<!--\; -\; -->1& 1\end{array}\right]}^T)$
for ${\mathrm{\lambda <!--\; \lambda \; -->}}_2=-<!--\; -\; -->1$.
(For ${\mathrm{\lambda <!--\; \lambda \; -->}}_3=6$ there are no eigenvectors).
Can the above eigenvectors and eigenvalues be used for solving this maximization problem?

In a book I'm reading, a claim is made that
$f(a)=r\cos <!--\; \; -->(a)-<!--\; -\; -->m\sin <!--\; \; -->(a)$
has the maximum
$\sqrt{r^2+m^2}$
(where a, r and m are real numbers).
But I'm not sure how to prove it or even if it's true. Setting ${\displaystyle \frac{d}{da}}f(a)=0$ gives $-<!--\; -\; -->r\sin <!--\; \; -->(a)-<!--\; -\; -->m\cos <!--\; \; -->(a)=0$ or $\tan <!--\; \; -->(a)=-<!--\; -\; -->m/r$, so $a=\arctan <!--\; \; -->(-<!--\; -\; -->m/r)$. Plugging that in into wolfram does not give the claimed result...

$x^{2}y\to <!--\; \to \; -->$ max,
such that $x^2+4xy\le <!--\; \le \; -->1,x\ge <!--\; \ge \; -->0$ and $y\ge <!--\; \ge \; -->0$.

I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone please give me an example of how this should be done? And should I include the constraints $x\ge <!--\; \ge \; -->0$ and $y\ge <!--\; \ge \; -->0$ into the Lagrangian function?

Rotate vector by a random little amount
Suppose there is a vector $v=(v_x,v_y,v_z)$
I want to rotate this vector by a random little amount (let's say at most $10º$ ). How can I do that?

I am maximizing
$f(x,y)=-<!--\; -\; -->x$
given the constraint
$g(x,y)=x^2-<!--\; -\; -->y^2=0$
To satisfy the non degenerate constraint qualification I have:
$Dg(x,y)=[2x-<!--\; -\; -->2y]$
and the set of $(x,y)$ that satisfy it is having $x=y$.

However on setting up the Lagrange multiplier:
$L(x,y,\mathrm{\lambda <!--\; \lambda \; -->})=-<!--\; -\; -->x+\mathrm{\lambda <!--\; \lambda \; -->}(x^2-<!--\; -\; -->y^2)$
and getting the first order conditions:
$L_x=-<!--\; -\; -->1+2\mathrm{\lambda <!--\; \lambda \; -->}x=0$ and
$L_y=-<!--\; -\; -->2\mathrm{\lambda <!--\; \lambda \; -->}y=0$
$L_{\mathrm{\lambda <!--\; \lambda \; -->}}=x^2-<!--\; -\; -->y^2=0$
I have a contradiction since for $x=y$

The first equation will give:
$-<!--\; -\; -->2\mathrm{\lambda <!--\; \lambda \; -->}x=-<!--\; -\; -->1$
The second however shows:
$-<!--\; -\; -->2\mathrm{\lambda <!--\; \lambda \; -->}x=0$
Is there anywhere I have gotten wrong here?

Where does the Pythagorean theorem "fit" within modern mathematics?
I am interested in how today's professional mathematicians view the Pythagorean theorem, in terms of how the theorem fits within the axiomatic framework of mathematics. I often come across textbooks that define length by the Pythagorean theorem, so that the theorem is in essence a definition or axiom. In more modern mathematics such as linear algebra, is the Pythagorean theorem generally just used as the definition of length? Is it more conventional today to treat the Pythagorean theorem as a definition (or axiom) rather than a theorem? Are there any modern proofs of the Pythagorean theorem that don't rely on Euclidean geometry (like a proof that utilizes linear algebra/the dot product, etc.)?

Consider the function $r=f(\mathrm{\theta <!--\; \theta \; -->})$ in polar coordinates. The length of an arc of a circle is just
$S=\mathrm{\theta <!--\; \theta \; -->}r$
Where $r$ is the radius of the circle and $\mathrm{\theta <!--\; \theta \; -->}$ is the angle that represents this arc. But since $r=f(\mathrm{\theta <!--\; \theta \; -->})$ and $\mathrm{\theta <!--\; \theta \; -->}$ Should approach zero so that we can get the exact value of the arc, So
$dS=f(\mathrm{\theta <!--\; \theta \; -->})d\mathrm{\theta <!--\; \theta \; -->}$
Integrating from ${\mathrm{\theta <!--\; \theta \; -->}}_1$ to ${\mathrm{\theta <!--\; \theta \; -->}}_2$, we get :
$S={\int <!--\; \int \; -->}_{{\mathrm{\theta <!--\; \theta \; -->}}_1}^{{\mathrm{\theta <!--\; \theta \; -->}}_2}f(\mathrm{\theta <!--\; \theta \; -->})d\mathrm{\theta <!--\; \theta \; -->}$
But the actual formula for the length of a curve in polar coordinates is
${\int <!--\; \int \; -->}_{{\mathrm{\theta <!--\; \theta \; -->}}_1}^{{\mathrm{\theta <!--\; \theta \; -->}}_2}\sqrt{f^2(\mathrm{\theta <!--\; \theta \; -->})+f^{\prime }(\mathrm{\theta <!--\; \theta \; -->}{)}^2}d\mathrm{\theta <!--\; \theta \; -->}.$
I know that my approach isn’t rigorous enough, but it’s is still reasonable, so why it is different from the actual formula?

I have a multivariable function that I have defined the cost function and its gradient with respect to the variable vector. Let's call the cost function $f(\stackrel{\to <!--\; \to \; -->}{x})$, and variable vector $\stackrel{\to <!--\; \to \; -->}{x}$. I have been using nonlinear conjugate gradient descenet method for minimization. Algorithm is as follows:
$$\begin{gathered} k=0 \\ x=0 \\ {g}_0={\mathrm{\nabla <!--\; \nabla \; -->}}_{\stackrel{\to <!--\; \to \; -->}{x}}f(x_0) \\ \mathrm{\Delta <!--\; \Delta \; -->}{x}_0=-<!--\; -\; -->g_0 \end{gathered}$$
$$\begin{gathered} x_{k+1}\leftarrow <!--\; \leftarrow \; -->x_k+t\mathrm{\Delta <!--\; \Delta \; -->}{x}_k \\ {g}_{k+1}\leftarrow <!--\; \leftarrow \; -->\mathrm{\nabla <!--\; \nabla \; -->}f(x_{k+1}) \\ \mathrm{\Delta <!--\; \Delta \; -->}{x}_{k+1}\leftarrow <!--\; \leftarrow \; -->-<!--\; -\; -->g_{k+1}+\mathrm{\gamma <!--\; \gamma \; -->}\mathrm{\Delta <!--\; \Delta \; -->}{x}_k \end{gathered}$$
I know that I need to update formula so the solution "ascends" instead of descension. How can I update the algorithm? Also I wonder how the wolfe condition changes for maximization problems. Thank you and have a nice day.

I am very new to optimization andI have to solve this equation

max U(k)=tlog(1+ yhpg/(pg+s))+mpge^(-ky)) st k>0

can anyybody give me idea where should I start

I have an arc of a circle, and I have some other point in space (this might lie on the arc or it might not). I am looking for a formula that will compute the closest point on the arc to the other point. I also need to be able to get the distance from the start of the arc to this closest point.
what I know about the arc is: (1) the circle centre, (2) the angle of the beginning of the arc, (3) the arc angle, (4) the circle radius, (5) begin and end points of the arc on the circle circumference, (6) the direction the arc is going (i.e. Clockwise or Counter-clockwise)
what I know about the other point is: (1) its position

I know how to solve maximization problems on numbers, and I know how to solve differential equations which are equations on functions, but how do I solve a 'maximization problem' over functions?
Here is a specific problem:
Find a positive real function $F(x)$, continuous and monotonically increasing on the real interval $[0,1]$, which maximizes:
$\frac{F(x)}{F(1)}$
Subject to:
$F^{\prime }(x)=(1-<!--\; -\; -->x)F^{″}(x)$
what is the function $F$ which attains this maximum?

let $T\ge <!--\; \ge \; -->$1 be some finite integer, solve the following maximization problem.
Maximize $\underset{t=1}{\overset{T}{\sum <!--\; \sum \; -->}}$($\frac{1}{2}$)$\sqrt{x_t}$ subject to $\underset{t=1}{\overset{T}{\sum <!--\; \sum \; -->}}$, $x_t\le <!--\; \le \; -->1$, $x_t\ge <!--\; \ge \; -->0$, t=1,...,T
I have never had to maximize summations before and I do not know how to do so. Can someone show me a step by step break down of the solution?

Point C moves along the top arc of a circle of radius 1 centered at the origin O(0, 0) from point A(-1, 0) to point B(1, 0) such that the angle BOC decreases at a constant rate 1 radian per minute. How does the area of the triangle ABC change at the moment when |AC|=1? Answer: it increases at 1/2 square units per minute. Could you give me a hint how to solve this task? I don't even know what to begin with.