Question

The number of solutions(s) of the equation $3\tan x+x^3=2\forall x\in \left(0,\frac{\pi }{4}\right)$ is .

Answer 1

Let f(x) = $3\tan x+x^3-2$$\mathrm{Thus},f\text{'}\left(x\right)=3{\sec }^{2}x+3x^2>0.\mathrm{Hence},f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{function}.\mathrm{Also},f\left(0^{+}\right)=-2<0\mathrm{and}f\left(\frac{\pi }{4}\right)>0.\mathrm{So},\mathrm{by}\mathrm{Intermediate}\mathrm{Value}\mathrm{Theorem},f\left(x\right)=0\mathrm{has}\mathrm{exactly}\mathrm{one}\mathrm{root}\mathrm{in}\left(0,\frac{\pi }{4}\right)$