~Find the local maxima and local minima, of the function f(x) = sin x – cos x, 0 < x < 2pi.
Also find the local maximum and local minimum values.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{sinx}-\mathrm{cosx} \mathrm{x}\in \left(0, 2\mathrm{\pi }\right) \\ \Rightarrow \mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{cosx}-\left(-\mathrm{sinx}\right)=\mathrm{cosx}+\mathrm{sinx} \\ \mathrm{For} \mathrm{maxima} \mathrm{or} \mathrm{minima}, \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=0 \\ \Rightarrow \mathrm{cosx}+\mathrm{sinx}=0 \\ \Rightarrow \mathrm{cosx}=-\mathrm{sinx} \\ \Rightarrow \frac{\mathrm{sinx}}{\mathrm{cosx}}=-1 \\ \Rightarrow \mathrm{tanx}=-1 \\ \mathrm{x}=\frac{3\mathrm{\pi }}{4} \mathrm{or} \mathrm{x}=\frac{7\mathrm{\pi }}{4} \\ \mathrm{Again} \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\mathrm{cosx}+\mathrm{sinx} \\ \Rightarrow \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)=-\mathrm{sinx}+\mathrm{cosx} \\ \Rightarrow \mathrm{f}\text{'}\text{'}\left(\frac{3\mathrm{\pi }}{4}\right)=-\sin \frac{3\mathrm{\pi }}{4}+\cos \frac{3\mathrm{\pi }}{4}=-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}=-\sqrt{2} \mathrm{which} \mathrm{is} \mathrm{negative}. \\ \mathbf{Hence}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{has}\mathbf{ }\mathbf{local}\mathbf{ }\mathbf{maxima}\mathbf{ }\mathbf{at}\mathbf{ }\mathbf{x}\mathbf{=}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{4}}\mathbf{.} \\ \Rightarrow \mathrm{f}\text{'}\text{'}\left(\frac{7\mathrm{\pi }}{4}\right)=-\sin \frac{7\mathrm{\pi }}{4}+\cos \frac{7\mathrm{\pi }}{4}=-\left(-\frac{1}{\sqrt{2}}\right)+\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2} \mathrm{which} \mathrm{is} \mathrm{positive}. \\ \mathbf{Hence}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{has}\mathbf{ }\mathbf{local}\mathbf{ }\mathbf{minima}\mathbf{ }\mathbf{at}\mathbf{ }\mathbf{x}\mathbf{=}\frac{\mathbf{7}\mathbf{\pi }}{\mathbf{4}}\mathbf{.} \end{gathered}$$

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