Min value of a trigonometric expression [closed]

Hik Aubergine 10/16/2018. 3 answers, 261 views
calculus optimization

What is the value of $\sin(x)$ for the maximum value of $(5+3\sin(x))^2 (7-3\sin(x))^3$.

3 Answers

lab bhattacharjee 10/16/2018.


AM GM inequality

$$\dfrac{2\cdot3(5+3\sin x)+3\cdot2(7-3\sin x)}{2+3}\ge\sqrt[5]{3^2(5+3\sin x)^22^3(7-3\sin x)^3}$$

The equality occurs if $3(5+3\sin x)=2(7-3\sin x)$

karakfa 10/16/2018.

AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...

let $z=\sin(x)$, $f(z)=g(z)^2h(z)^3$

$$ f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh') $$

since $-1 \le z \le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.

gt6989b 10/16/2018.


Let $z = \sin x$. You are asking for what value of $z$ does the function $$f(z) = (5+3z)^2 (7-3z)^3$$ attains its maximum. Taking a derivative would likely help.


As mentioned in the comments below, you are only optimizing over $-1 \le z \le 1$.

HighResolutionMusic.com - Download Hi-Res Songs

1 (G)I-DLE


(G)I-DLE. 2018. Writer: Riot Music Team;Harloe.
2 The Chainsmokers

Beach House flac

The Chainsmokers. 2018. Writer: Andrew Taggart.
3 Ariana Grande

​Thank U, Next flac

Ariana Grande. 2018. Writer: Crazy Mike;Scootie;Victoria Monét;Tayla Parx;TBHits;Ariana Grande.
4 Nicki Minaj

No Candle No Light flac

Nicki Minaj. 2018. Writer: Denisia “Blu June” Andrews;Kathryn Ostenberg;Brittany "Chi" Coney;Brian Lee;TJ Routon;Tushar Apte;ZAYN;Nicki Minaj.
5 Clean Bandit

Baby flac

Clean Bandit. 2018. Writer: Jack Patterson;Kamille;Jason Evigan;Matthew Knott;Marina;Luis Fonsi.
6 Imagine Dragons

Bad Liar flac

Imagine Dragons. 2018. Writer: Jorgen Odegard;Daniel Platzman;Ben McKee;Wayne Sermon;Aja Volkman;Dan Reynolds.
7 Halsey

Without Me flac

Halsey. 2018. Writer: Halsey;Delacey;Louis Bell;Amy Allen;Justin Timberlake;Timbaland;Scott Storch.

Waste It On Me flac

BTS. 2018. Writer: Steve Aoki;Jeff Halavacs;Ryan Ogren;Michael Gazzo;Nate Cyphert;Sean Foreman;RM.
9 BlackPink

Kiss And Make Up flac

BlackPink. 2018. Writer: Soke;Kny Factory;Billboard;Chelcee Grimes;Teddy Park;Marc Vincent;Dua Lipa.
10 Fitz And The Tantrums

HandClap flac

Fitz And The Tantrums. 2017. Writer: Fitz And The Tantrums;Eric Frederic;Sam Hollander.
11 Backstreet Boys

Chances flac

Backstreet Boys. 2018.
12 Kelly Clarkson

Never Enough flac

Kelly Clarkson. 2018. Writer: Benj Pasek;Justin Paul.
13 Diplo

Close To Me flac

Diplo. 2018. Writer: Ellie Goulding;Savan Kotecha;Peter Svensson;Ilya;Swae Lee;Diplo.
14 Anne-Marie

Rewrite The Stars flac

Anne-Marie. 2018. Writer: Benj Pasek;Justin Paul.
15 Little Mix

Woman Like Me flac

Little Mix. 2018. Writer: Nicki Minaj;Steve Mac;Ed Sheeran;Jess Glynne.
16 Imagine Dragons

Machine flac

Imagine Dragons. 2018. Writer: Wayne Sermon;Daniel Platzman;Dan Reynolds;Ben McKee;Alex Da Kid.
17 Little Mix

The Cure flac

Little Mix. 2018.
18 Bradley Cooper

Always Remember Us This Way flac

Bradley Cooper. 2018. Writer: Lady Gaga;Dave Cobb.
19 Rita Ora

Velvet Rope flac

Rita Ora. 2018.
20 Lady Gaga

I'll Never Love Again flac

Lady Gaga. 2018. Writer: Benjamin Rice;Lady Gaga.

Related questions

Hot questions


Popular Tags