 # High school geometry questions and answers

Recent questions in High school geometry Bernard Mora 2022-05-07 Answered

### In the one-variable case we can take a $\mathrm{log}$ transform of the function $f\left(x\right)\to \mathrm{log}\left(f\left(x\right)\right)$ and know that the same value maximizes both because log is increasing. The log turns products into sums which facilitates taking derivatives. Can the same argument be applied for multivariable cases? The question is: for a function $f\left(x,y\right)$ if I wish to find the ${x}^{\star },{y}^{\star }$ that maximize $f\left(x,y\right)$ are these the same ${x}^{\star \star },{y}^{\star \star }$ that maximize $\mathrm{log}\left(f\left(x,y\right)\right)$? My intuition is yes, nothing should be different in this multivariable case, but I desire a quick sanity check before setting off with computations. I would appreciate a rigorous proof or a counterexample. vilitatelp014 2022-05-07 Answered

### I have not yet had the privilege of studying multivariable calculus, but I have made an educated guess about how to find the minimum or maximum of a function with two variables, for example, $x$ and $y$.Since, in three dimensions, a minimum or maximum would be represented by a tangent plane with no slope in any direction, could I treat $y$ as a constant and differentiate $z$ with respect to $x$, then treat $x$ as a constant and differentiate with respect to $y$, and find the places where both of these two are equal to zero?Sorry if this is just a stupid assumption... it may be one of those things that just seems correct but is actually wrong. veleumnihryz 2022-05-03 Answered

### Let$f\left(z\right)=\frac{z\left(z-a\right)}{\left(z-b\right)\left(z-c\right)\left(z-d\right)}$be a complex rational function and let $S=\left\{z\in \mathbb{C}:|z|=1\right\}$ be the unit circle.My question:For which (non-zero complex numbers) $a,b,c$ and $d$ there exist an arc $C$ of $S$ such that $f\left(C\right)\subseteq S$ ? Bronson Olson 2022-05-03 Answered

### Maximize the value of the function$z=\frac{ab+c}{a+b+c},$where $a,b,c$ are natural numbers and are all lesser than 2010 and not necessarily distinct from each other. Please provide a proof, and if possible a general technique. Thank you. micelije1mw 2022-05-03 Answered

### I have encountered that in some cases maximization of a function had been substituted with a maximization of its monotone transformation.For example, finding the min or max of $f\left(x,y\right)=\left(\left(x-1{\right)}^{2}+\left(y-2{\right)}^{2}{\right)}^{1/2}$ is the same as finding the min / max of $\left({x}^{2}+{y}^{2}\right)$ and then take the square root of the answer. So, my questions are:1) Is it true for any monotone transformation $g$?2) Is the same principle allow us to maximize /minimize $\mathrm{log}g\left(x\right)$ instead of $g\left(x\right)$ itself? Beedgighref28n 2022-05-03 Answered

### I have non-negative function $g\left(y,x\right)$ that is define using non-negative $f\left(y,x\right)$ in the following way:$\begin{array}{r}g\left(y,x\right)={\int }_{0}^{x}f\left(t,y\right)dt\end{array}$I am trying to maximize $\underset{y\in S}{max}g\left(y,x\right)$. Using Fatou's lemma we have that$\begin{array}{r}\underset{y\in S}{max}g\left(y,x\right)={\int }_{0}^{x}f\left(t,y\right)dt\le {\int }_{0}^{x}\underset{y\in S}{max}f\left(t,y\right)dt\end{array}$I also have that $\underset{y\in S}{max}f\left(t,y\right)\le h\left(t\right)$ where ${\int }_{0}^{x}h\left(t\right)<\mathrm{\infty }$My question when does the last inequality hold with equally?What do I have to assume about $f\left(y,x\right)$ and $g\left(x,y\right)$ for equality to hold?Can I apply dominated convergence theorem here? Abraham Bradley 2022-05-03 Answered

### Pythagorean theorem does require the Cauchy-Schwarz inequality?In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved by the Cauchy-Schwarz inequality? Or effectively is the Pythagorean Theorem a stronger if not equivalent notion? ngihlungeqtr 2022-05-03 Answered

### This is a utility maximzation problemmaximize ${x}^{a}+{y}^{b}$ subject to ${p}_{1}x+{p}_{2}y=w$ (utility maximization problem)Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see it. many thanks!!! NepanitaNesg3a 2022-05-02 Answered

### I maximized the function$f\left(x,y\right)=\left(x+2\right)\left(y+1\right)$subject to$4x+6y=130,\phantom{\rule{1em}{0ex}}x,y\ge 0$and I found $\left(x,y\right)=\left(16,11\right)$. But the question I'm doing asks me to analyse the "sufficient conditions of second order".So, I thought I should use some theorem that tells me (16,11) is the max value of $f$. I don't know if I could use Weierstrass' Theorem here and I believe the hessian matrix doesn't help me either. estroishave4 2022-05-02 Answered

### Maximize${x}_{2}-{x}_{1}+{y}_{1}-{y}_{2}$given that ${x}_{1}^{2}+{y}_{1}^{2}=1$ and ${x}_{2}^{2}+{y}_{2}^{2}=1$.I was thinking about using Lagrange multipliers, but I only know how that works for a 3-variable function, not 4. Could someone please suggest a way to solve this? Maybe with Lagrange multipliers or some more elementary method? gaitaprepeted05u 2022-05-01 Answered

### How to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle. hafusxl 2022-05-01 Answered

### Suppose I have $n$ integers ${\mu }_{i}$, $i=\left\{1,2,...,n\right\}$. Define $\overline{\mu }=\frac{1}{n}\sum _{i=1}^{n}{\mu }_{i}$. It is given that all the ${\mu }_{i}$'s are either +1 or −1. How can I show that $\sum _{i=1}^{n}\left({\mu }_{i}-\overline{\mu }{\right)}^{2}$ is maximized only when $\frac{⌊n⌋}{2}$ or $\frac{⌈n⌉}{2}$ are +1 and rest are −1's.I was thinking about proving this inductively, but I am not sure how. Essence Byrd 2022-05-01 Answered

### If $X\sim$ Poisson $\left(\lambda \right)$, prove that $P\left[X=k\right]$ is maximized at $k=\lambda$How would I go about proving something like this so that the method would work to prove a maximization point of other random variables like say Binomial is max at $k=\left(n+1\right)\star p$? nrgiizr0ib6 2022-04-30 Answered

### If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle? kabutjv7 2022-04-30 Answered

### Prove that the product of two lines equations is hyperbola komanijuuxw 2022-04-29 Answered

### Let L :$:{P}_{2}⇒{P}_{3}$ be a linear transformation for which weknow that $L:\left(1\right)=1,L\left(t\right)={t}^{2},L\left({t}^{2}\right)={t}^{3}=t.$(a) Find $L\left(2{t}^{2}-5t=3\right).$(b) Find $L\left(a{t}^{2}-bt+c\right).$ Bruce Rosario 2022-04-22 Answered

### How to find the coefficient a of a$y=a{x}^{2}$parabola? Yaritza Robinson 2022-04-21 Answered

### Directions: Use cross-product property proportion to verify that the ratio is equal. The first one is done for you.PLEASE… Answer it correctly with solutions. Thank You!  Lexi Chandler 2022-04-07 Answered