Question

Find the local maxima and local minima of the function $f\left(x\right)=sinx-cosx,0<x<2\pi$. Also find the local maximum and local minimum values.

Answer 1

Given $f\left(x\right)=\sin x-\cos x$

differentiate on both sides

$f^{{}^{\prime }}\left(x\right)=\sin x+\cos x$

$f^{{}^{\prime }}\left(x\right)=0\Rightarrow x=\frac{3\pi }{4},\frac{7\pi }{4}$

local maxima:

$f^{{}^{\prime \prime }}\left(x\right)=\cos x-\sin x$

$f^{{}^{\prime \prime }}\left(\frac{3\pi }{4}\right)=-\sqrt{2}<0,$

$f\left(\frac{3\pi }{4}\right)=\sqrt{2}$

local minima:

$f^{{}^{\prime \prime }}\left(\frac{7\pi }{4}\right)=\sqrt{2}>0,f\left(\frac{3\pi }{4}\right)=-\sqrt{2}$

$\sqrt{2}$ and $-\sqrt{2}$ are the maximum and minimum values.