If g(x)=f(\tan^2 x−2\tan x+4) , \ 0<x<\frac{\pi}{2}, then g(x) is increasing in what interval

Jazmin Perry 2022-01-24 Answered
If g(x)=f(tan2x2tanx+4), 0<x<π2, then g(x) is increasing in what interval?
I have first differentiated the given equation and found the value of x for which the interior part of f is 3 ,this gives x=π4 and it gives me g(π4)=0 but I am stuck here and don't know what to do after this.
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Expert Answer

Jason Duke
Answered 2022-01-25 Author has 11 answers
f′′(x)>0 so f′ increasing. Therefore: x<3f(x)<f(3)f(x)<0x<3
g(x)=f(tan2x2tanx+4)2tanx2cos2x
g(x)=0tanx=1x=π4
Therefore for x(0,π4):2tanx2<0g(x)>0 so g increasing on (0,π4) for x(π4,π2):2tanx2>0g(x)<0 so g decreasing on (π4,π2)
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