Find the least value for sin⁡x−cos2x−1At first I found the first derivative to be y′=cos⁡x+2sin⁡xcos⁡xCritical...

Advadlippabrj

Advadlippabrj

Answered

2022-01-27

Find the least value for sinxcos2x1
At first I found the first derivative to be y=cosx+2sinxcosx
Critical point 0,π6 (principal)
ysinx+2(cos2x)
Then substitution of x by critical points I found minima. But my answer is incorrect.
Correct minimum value is 94

Answer & Explanation

Aiden Cooper

Aiden Cooper

Expert

2022-01-28Added 14 answers

f(x)=sin2x+sin{x}2=(sin{x}+12)22.252.25
The equality occurs for sinx=12, which says that -2.25 is a minimal value.
Micah May

Micah May

Expert

2022-01-29Added 11 answers

Yes your way is correct indeed for x=π6 we have
f(x)=12341=94
As an alternative, recall that cos2x=1sin2x and therefore we have
f(x)=sinxcos2x1=sin2x+sinx2
then set t=sinx and consider the parabola
g(t)=t2+t2g(t)=2t+1=0t=12

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