Find the minimum value of the function \frac{\sin 5x}{\sin^5 x}

David Troyer

David Troyer

Answered question

2022-01-17

Find the minimum value of the function sin5xsin5x

Answer & Explanation

Ethan Sanders

Ethan Sanders

Beginner2022-01-18Added 35 answers

We have:
sin(5x)sin5x=16sin5x20sin3x+5sinxsin5x=1620sin2x+5sin4x=1620t+5t2=f(t),.
t=1sin2x1
You can take it from here as you have a quadratic function in t1
scoollato7o

scoollato7o

Beginner2022-01-19Added 26 answers

hint
The derivative is
5cos5xsinxcosxsin5xsin10xsin4x
=5sin4xsin6x
Your function is 2π−periodic, even and satisfies
f(πx)=f(x)
So, you just need to find the minimum at [0,π2]
alenahelenash

alenahelenash

Expert2022-01-24Added 556 answers

sin5x=R(eix)5=5cos4xsinx10cos2xsin3x+sin5x=16sin5x20sin3x+5sinx

and we want to maximize
1620t1+5t2
over [0,1], where t:=sin2x
By canceling the derivative, 20t210t3
gives the root t=12 and the minimum value 16202+54=4

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