Question

Show that f(x) = x² − x sin x is an increasing function on (0, π/2).

Answer 1

$$\begin{gathered} f\left(x\right)=x^2-x\sin x \\ f\text{'}\left(x\right)=2x-x\cos x-\sin x \\ \mathrm{Here}, \\ 0<x<\frac{\mathrm{\pi }}{2} \\ \Rightarrow 0<\sin x<1\text{and}0<\cos x<1 \\ \Rightarrow 2x-x\cos x-\sin x>0 \\ \Rightarrow f\text{'}\left(x\right)>0,\forall x\in \left(0,\frac{\mathrm{\pi }}{2}\right) \\ \text{So,}f\left(x\right)\text{is increasing on}\left(0,\frac{\mathrm{\pi }}{2}\right)\text{.} \end{gathered}$$