Prove This beautiful result !

$\alpha \quad ,\quad \beta \quad >\quad 0\quad \quad \& \quad \quad \theta \quad \in \quad R\\ \\ f\left( \theta \right) \quad =\quad \quad \alpha \sec ^{ 2 }{ \theta } \quad +\quad \beta \csc ^{ 2 }{ \theta } \quad +\quad \sqrt { { \alpha }^{ 2 }\sec ^{ 4 }{ \theta } \quad +\quad { \beta }^{ 2 }\csc ^{ 4 }{ \theta } }$ .

Then Prove That Minimum Value of $f\left( \theta \right)$ is :

${ f\left( \theta \right) }_{ min }\quad =\quad 2(\alpha \quad +\quad \beta \quad +\quad \sqrt { 2\alpha \beta } )$ .

Extra credit: Just observe it !
( It's Look ugly But Believe me in actual it is very very Beautiful question )

Following tool of mathematics are may Helpful Here Like : Substitution , Geometrical interpretation , AM - GM , Circles , Straight lines , Polar substitution etc, !!

#Geometry #Minimum #GeometricInterpretation #Beauty-of-Maths

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Sort by:

Top Newest

Okay Now its time to give Hint :

use Substitution : $a\quad =\quad \alpha \sec ^{ 2 }{ \theta } \\ \\ b\quad =\quad \beta \csc ^{ 2 }{ \theta }$ .

Deepanshu Gupta - 6 years, 4 months ago

Ohh ! Now I got something

is it mean :

$f(\theta )\quad =\quad a\quad +\quad b\quad +\quad \sqrt { { a }^{ 2 }\quad +\quad { b }^{ 2 } }$ .

I think it is Perimeter of right angle triangle ?

Am i going Correct ?

Karan Shekhawat - 6 years, 4 months ago

Yes ! You are on right Path ! Keep Trying You will find the way :)

Deepanshu Gupta - 6 years, 4 months ago

good Question,,!! rememberedme to high school..!! :D

Wijil Pambudi - 6 years, 4 months ago

See Unravelling an inequality problem by Calvin Lin.

Souryajit Roy - 6 years, 4 months ago

That's certainly one way to motivate the solution.

Recognizing and relating patterns is helpful in approaching such problems.

Calvin Lin Staff - 6 years, 4 months ago

@Calvin Lin Sir I solved in this way :(Similar to your Awesome Technique )

imgur imgur

Let $P\quad =\quad a\quad +\quad b\quad +\quad \sqrt { { a }^{ 2 }\quad +\quad { b }^{ 2 } }$ .

also Let $\sqrt { { a }^{ 2 }\quad +\quad { b }^{ 2 } } \quad =\quad { C }_{ 1 }\quad +\quad { C }_{ 2 }$ .

$P\quad =\quad (\quad a\quad +\quad { C }_{ 2 }\quad )\quad +\quad (\quad b\quad +\quad { C }_{ 1 }\quad )\quad$ .

Now For Minimum perimeter P we use AM-GM inequality

$P\quad \ge \quad 2\sqrt { (\quad a\quad +\quad { C }_{ 2 }\quad )(\quad b\quad +\quad { C }_{ 1 }\quad ) } \\ \\ { P }_{ min }\quad =\quad 2\sqrt { (\quad a\quad +\quad { C }_{ 2 }\quad )(\quad b\quad +\quad { C }_{ 1 }\quad ) } \quad \\$

If and only iff :

1)- GM must be constant

2)- Equality of variable must be attained

$\\ \bullet \quad (\quad a\quad +\quad { C }_{ 2 }\quad )(\quad b\quad +\quad { C }_{ 1 }\quad )\quad =\quad constant\\ \\ \bullet \quad (\quad a\quad +\quad { C }_{ 2 }\quad )\quad =\quad (\quad b\quad +\quad { C }_{ 1 }\quad )\quad =\quad r\quad (say)\\ \\ { P }_{ min }\quad =\quad 2r\quad$ .

Now Same as Yours !

Is This is correct ? Calvin Sir ?

Deepanshu Gupta - 6 years, 4 months ago

ohh is it a famous problem? I don't know about this . Actually this question was giving to me by a friend of me as a challange !

So I Liked it after solving it So i share this ! ,

But i solved it in slightly different manner as that calvin sir posted ( also my friend also help me little in how to starting this problem )

Deepanshu Gupta - 6 years, 4 months ago

So what is your Slightly different approach Can you post it ! please Thanks ! @Deepanshu Gupta

Karan Shekhawat - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$