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

If , 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 $x=\frac{\pi }{4}$ and it gives me $g\left(\frac{\pi }{4}\right)=0$ but I am stuck here and don't know what to do after this.
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Jason Duke
f′′(x)>0 so f′ increasing. Therefore: $x<3⇔f\prime \left(x\right)
$g\prime \left(x\right)=f\prime \left({\mathrm{tan}}^{2}x-2\mathrm{tan}x+4\right)\cdot \frac{2\mathrm{tan}x-2}{{\mathrm{cos}}^{2}x}$
$g\prime \left(x\right)=0⇔\mathrm{tan}x=1⇔x=\frac{\pi }{4}$
Therefore for $x\in \left(0,\frac{\pi }{4}\right):2\mathrm{tan}x-2<0⇔g\prime \left(x\right)>0$ so g increasing on $\left(0,\frac{\pi }{4}\right)$ for $x\in \left(\frac{\pi }{4},\frac{\pi }{2}\right):2\mathrm{tan}x-2>0⇔g\prime \left(x\right)<0$ so g decreasing on $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$