Concavity of a function

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k a March Highlights and 2025 AoPS Online Class Information

jlacosta 0

Mar 2, 2025

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0 replies

jlacosta

Mar 2, 2025

0 replies

k i Suggestion Form

jwelsh 0

May 6, 2021

Hello!

Given the number of suggestions we’ve been receiving, we’re transitioning to a suggestion form. If you have a suggestion for the AoPS website, please submit the Google Form:
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To keep all new suggestions together, any new suggestion threads posted will be deleted.

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Please check the FTW! thread for bugs and post any new ones in the For the Win! and Other Games Support Forum.

0 replies

jwelsh

May 6, 2021

0 replies

k i Read me first / How to write a bug report

slester 3

N May 4, 2019 by LauraZed

Greetings, AoPS users!

If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.

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Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.

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...
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Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.

How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.

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It'll look something like this:
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3 replies

slester

Apr 9, 2015

LauraZed

May 4, 2019

k i Community Safety

dcouchman 0

Jan 18, 2018

If you find content on the AoPS Community that makes you concerned for a user's health or safety, please alert AoPS Administrators using the report button (Z) or by emailing sheriff@aops.com . You should provide a description of the content and a link in your message. If it's an emergency, call 911 or whatever the local emergency services are in your country.

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0 replies

dcouchman

Jan 18, 2018

0 replies

Differentiation Marathon!

LawofCosine 182

N 39 minutes ago by LawofCosine

Hello, everybody!

This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!

Have fun!

(Sorry about the bad english)

182 replies

LawofCosine

Feb 1, 2025

LawofCosine

39 minutes ago

Integration Bee Kaizo

Calcul8er 45

N 2 hours ago by Calcul8er

Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!

45 replies

Calcul8er

Mar 2, 2025

Calcul8er

2 hours ago

Double factorial identity

Snoop76 0

Yesterday at 6:31 PM

Source: own

Show that the following identity holds: $\sum_{k=0}^n (2k+3)!!{n\choose k}=2(n+1)\sum_{k=0}^n (2k+1)!!{n\choose k}+\sum_{k=0}^n (2k-1)!!{n\choose k}$

0 replies

Snoop76

Yesterday at 6:31 PM

0 replies

Concavity of a function

pii-oner 1

N Yesterday at 1:50 PM by alexheinis

Hi everyone,

I am studying the concavity of the function

$f(x) = \sqrt{1 - x^a}, \quad a \geq 0$
on the interval $x \in [0,1]$ .

I computed the second derivative and found that for $a \geq 1$ , the function appears to be concave. However, I am uncertain about the behavior at the endpoints.

Does anyone have insights on confirming concavity rigorously for $a \geq 1$ and understanding the behavior at the endpoints? Any help would be greatly appreciated!

Thanks!

1 reply

pii-oner

Yesterday at 9:00 AM

alexheinis

Yesterday at 1:50 PM

k 2023 IMO

ostriches88 4

N Mar 14, 2025 by jlacosta

As outlined in this (locked) post, the forum/blog creation message is out of date. It has been over a year since that post, and it still has not changed. I don't know if this got lost somewhere in the "passing this along" chain or if it was determined to be irrelevant, but it seems like a relatively simple fix ;)

4 replies

ostriches88

Mar 10, 2025

jlacosta

Mar 14, 2025

Pressing 'go down button' always creates a gray box on the last post

Craftybutterfly 13

N Mar 14, 2025 by Craftybutterfly

Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post
Page URL: any forum
Steps to reproduce:
1. Go to any topic in a forum
2. The gray box at the bottom overlaps part of the first post
Expected behavior: Should not show a gray box
Frequency: 100% of the time
Operating system(s): Linux HP EliteBook 835 G8 Notebook PC
Browser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)
Additional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears

13 replies

Craftybutterfly

Mar 12, 2025

Craftybutterfly

Mar 14, 2025

k Somehow I broke the unique contest collection naming...

Equinox8 1

N Mar 14, 2025 by jlacosta

Somehow I broke the unique contest collection naming scheme; for what it's worth, I also ran into an AJAX timeout error while trying to make this particular post collection for the first time, and pressed "create" twice.

Not sure if there's anything to be done here, but are there potentially unforeseen consequences here?

https://artofproblemsolving.com/community/c4258765_2024_puerto_rico_team_selection_test (this is the one I'm working on; it's also not finished yet)
https://artofproblemsolving.com/community/c4258764_2024_puerto_rico_team_selection_test

1 reply

Equinox8

Mar 12, 2025

jlacosta

Mar 14, 2025

k New Forums Duplicates

k1glaucus 3

N Mar 14, 2025 by jlacosta

Summary of the problem: In the New Forums collection, many of the forums are duplicated, although some have slightly different information (likely due to the forum being edited by its admin(s)). Typically only one of the two has threads, and I believe this is the only way to access
Page URL: https://artofproblemsolving.com/community/c74_new_forums
Steps to reproduce:
1. Go to the link
2. You should see duplicates of some forums
Expected behavior: Each forum appears once
Frequency: Every time
Additional information: Typically only one of the two has threads, and from creating forums in the past I believe this is the only way to access the duplicate. Also not all of the forums are duplicated. Another reason to suspect these are duplicates are the forums with similar names have the same admin.

3 replies

k1glaucus

Mar 12, 2025

jlacosta

Mar 14, 2025

k Pi Day!!

SomeonecoolLovesMaths 66

N Mar 14, 2025 by LawofCosine

Happy $\pi$ day everyone!

66 replies

SomeonecoolLovesMaths

Mar 14, 2025

LawofCosine

Mar 14, 2025

k Help with beast academy

tyrantfire4 1

N Mar 14, 2025 by tyrantfire4

So my siblings do BA and it won't work if anyone can fix it that would be great!
All browers won't work
Apple and fire
Open BA account

1 reply

tyrantfire4

Mar 14, 2025

tyrantfire4

Mar 14, 2025

Search function in private messages not working

WisteriaV 2

N Mar 12, 2025 by mathlearner2357

For the past while, the search function in private messages hasn’t been working. Whenever I search for anything, it says, “No topics here!” after trying to load for a while. I’ve tried different devices (laptop and ipad) and browsers (chrome on both devices, safari on ipad, and microsoft edge on laptop), and the results are the same. I’ve also had friends say the same happens for them.

2 replies

WisteriaV

Mar 12, 2025

mathlearner2357

Mar 12, 2025

Viewing next classes

lilorocks11 2

N Mar 11, 2025 by jlacosta

Hello,

I'm wondering how to look at my next classes. I'm not entirely sure if I'm registered for a class. I tried looking around on the My AoPS page, but didn't find anything that had to do with the course I wanted.

2 replies

lilorocks11

Mar 8, 2025

jlacosta

Mar 11, 2025

Error when abandoning quest

dragonborn56 3

N Mar 11, 2025 by pb0975

I attempted to abandon the Alcumus quest Hit The Gym: Accuracy. I clicked on it and selected abandon, but the popup disappeared and the quest was still there. There were no changes to the log, and my daily abandon was used, as when I clicked the quest again it said that all of my abandons were used.

3 replies

dragonborn56

Mar 10, 2025

pb0975

Mar 11, 2025

What????

LawofCosine 21

N Mar 10, 2025 by LawofCosine

Umm....what is happening? The $-\infty$ seems to be squashed but the $\infty$ is normal. This is from the AoPS Calculus ebook.

21 replies

LawofCosine

Mar 7, 2025

LawofCosine

Mar 10, 2025

Concavity of a function

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3 posts

#1 h Yesterday at 9:00 AM

Y by

Hi everyone,

I am studying the concavity of the function

$f(x) = \sqrt{1 - x^a}, \quad a \geq 0$
on the interval $x \in [0,1]$ .

I computed the second derivative and found that for $a \geq 1$ , the function appears to be concave. However, I am uncertain about the behavior at the endpoints.

Does anyone have insights on confirming concavity rigorously for $a \geq 1$ and understanding the behavior at the endpoints? Any help would be greatly appreciated!

Thanks!

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

10471 posts

#2 h Yesterday at 1:50 PM

Y by

We have $f'(x)={{-ax^{a-1}}\over {2\sqrt{1-x^a}}}$ . Now $x^{a-1}$ increases if $a>1$ and $1-x^a$ decreases. Hence ${1\over {\sqrt{1-x^a}}}$ increases and $f$ is indeed concave.
The endpoints are not a problem: if $f$ is concave on $(a,b)$ and continuous on $[a,b]$ , then it is also concave on $[a,b]$ .
The proof is easy: suppose $c<b, \lambda,\mu\ge 0, \lambda+\mu=1$ . Also choose $c<d<b$ . Then $f(\lambda c+\mu d)\ge \lambda f(c)+\mu f(d)$ .
Now send $d\uparrow b$ to obtain $f(\lambda c+\mu b)\ge \lambda f(c)+\mu f(b)$ . Same thing when the left point is $x=a$ .
And welcome to AOPS.

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