Question

Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing.

Answer 1

$$\begin{gathered} f\left(x\right)=\sin x-\cos x,x\in \left(0,2\mathrm{\pi }\right) \\ f\text{'}\left(x\right)=\cos x+\sin x \\ \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be increasin, we must have} \\ f\text{'}\left(x\right)>0 \\ \Rightarrow \cos x+\sin x>0 \\ \Rightarrow \sin x>-\cos x \\ \Rightarrow \tan x>-1 \\ \Rightarrow x\in \left(0,\frac{3\mathrm{\pi }}{4}\right)\cup \left(\frac{7\mathrm{\pi }}{4},2\mathrm{\pi }\right) \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}\left(0,\frac{3\mathrm{\pi }}{4}\right)\cup \left(\frac{7\mathrm{\pi }}{4},2\mathrm{\pi }\right). \end{gathered}$$

$$\begin{gathered} \text{For}\mathit{\text{f}}\text{(}\mathit{\text{x}}\text{) to be decreasing we must have,} \\ f\text{'}\left(x\right)<0 \\ \Rightarrow \cos x+\sin x<0 \\ \Rightarrow \sin x<-\cos x \\ \Rightarrow \tan x<-1 \\ \Rightarrow x\in \left(\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right) \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is decreasing on}\left(\frac{3\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right)\text{.} \end{gathered}$$