Maximum

Geometry Level 4

$\large \dfrac{9 \sin(\omega)\cos(\omega)}{\sqrt{9 \sin^{2}(\omega)+16}}$ Find maximum value of above expression.

Problem is original.

The answer is 1.

2 solutions

Otto Bretscher
Nov 10, 2015

We observe that $72x-81x^2\leq 16$ for all $x$ ; equality holds when $x=\frac{4}{9}$ . Rearranging the terms, we find that $\frac{81x(1-x)}{9x+16}\leq 1$ for non-negative $x$ . Now we let $x=\sin^2\omega$ and take the square root to see that the maximum is $\boxed{1}$ .

Moderator note:

What is the motivation behind the observation that $72 x - 81 x^2 \leq 16$ ?

Again, I'm "working backwards", which is "bad form", as you told me in another problem. I will try to motivate my steps better in the future.

Otto Bretscher - 5 years, 6 months ago

You "worked backwards" from what, Comrade Otto?

Edit: Oh got it. Perfecting the square...

Pi Han Goh - 5 years, 6 months ago

@Otto Bretscher @Pi Han Goh Sir can you please explain me what you did??

Although I too solved the question but mine included lots of work on quadratics , like ranging the roots between $0$ and $1$ and many more things.

But if you could share your method , it will help me improve.

Ankit Kumar Jain - 4 years, 2 months ago

Marcos Lima
Nov 12, 2015

It follows immediately from Cauchy's inequality. One would notice that 16 =16sin^2(w) + 16cos^2(w) and the result is clear.

Can you explain your solution further? Thanks!

Calvin Lin Staff - 5 years, 6 months ago

Maximum

Problem is original.

The answer is 1.

2 solutions

Moderator note:

0 pending reports