Question

Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing.

Answer 1

$$\begin{gathered} f\left(x\right)=\sin x+\cos x,x\in \left[0,\frac{\mathrm{\pi }}{2}\right] \\ f\text{'}\left(x\right)=\cos x-\sin x \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasing, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow \cos x-\sin x>0 \\ \Rightarrow \sin x<\cos x \\ \Rightarrow \frac{\sin x}{\cos x}<1 \\ \Rightarrow \tan x<1 \\ \Rightarrow x\in [0,\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.) \end{gathered}$$