Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 75368 by vishalbhardwaj last updated on 10/Dec/19

$$$$$$\mathrm{Find}\mathrm{the}\mathrm{interval}\mathrm{for}\mathrm{which}\mathrm{the}\mathrm{function}$$$$f\left(x\right)=sinx+cosx,\mathrm{for}x\in [0,2\pi ]$$$$\mathrm{is}\mathrm{strictly}\mathrm{inceasing}\mathrm{and}\mathrm{srictly}\mathrm{decreasing}?$$

Commented by vishalbhardwaj last updated on 10/Dec/19

$$\mathrm{please}\mathrm{sir},\mathrm{write}\mathrm{complete}\mathrm{explanation}$$$$\mathrm{of}\mathrm{intervals}$$

Answered by MJS last updated on 10/Dec/19

$$f^{\prime }\left(x\right)=\cos x-\sin x$$$$f^{\prime }\left(x\right)=0$$$$\sin x=\cos x$$$$x=\frac{\pi }{4}\vee x=\frac{5\pi }{4}$$$$f^{″}\left(x\right)=-\sin x-\cos x$$$$f^{″}\left(\frac{\pi }{4}\right)<0\Rightarrow \mathrm{maximum}$$$$f^{″}\left(\frac{5\pi }{4}\right)>0\Rightarrow \mathrm{minimum}$$$$[0;\frac{\pi }{4}[f\left(x\right)\mathrm{strictly}\mathrm{increasing}$$$$]\frac{\pi }{4};\frac{5\pi }{4}[f\left(x\right)\mathrm{strictly}\mathrm{decreasing}$$$$]\frac{5\pi }{4};2\pi ]f\left(x\right)\mathrm{strictly}\mathrm{increasing}$$

Commented by vishalbhardwaj last updated on 10/Dec/19

$$\mathrm{sir}\mathrm{please}\mathrm{solve}\mathrm{by}\mathrm{first}\mathrm{derivative}\mathrm{rule}$$$$\mathrm{and}\mathrm{please}\mathrm{explane}\mathrm{by}\mathrm{taking}\mathrm{any}\mathrm{value}\mathrm{in}$$$$\mathrm{between}\mathrm{the}\mathrm{intervals}$$

Commented by MJS last updated on 10/Dec/19

$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{know}\mathrm{this}\mathrm{rule}.\mathrm{the}\mathrm{proper}\mathrm{way}\mathrm{I}\mathrm{used}$$$$\mathrm{is}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{zeros}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{derivate}\mathrm{to}\mathrm{get}$$$$\mathrm{extremes}\mathrm{and}\mathrm{then}\mathrm{check}\mathrm{the}\mathrm{second}\mathrm{derivate}$$$$\mathrm{to}\mathrm{find}\mathrm{if}\mathrm{these}\mathrm{are}\mathrm{maxima}\mathrm{or}\mathrm{minima}$$$$f^{\prime }\left(p\right)=0\wedge f^{″}\left(p\right)<0\iff \max \mathrm{at}p$$$$f^{\prime }\left(p\right)=0\wedge f^{″}\left(p\right)>0\iff \min \mathrm{at}p$$

Answered by Kunal12588 last updated on 10/Dec/19

$$f{}^{\prime }\left(x\right)=cosx-sinx=\sqrt{2}cos(\frac{\pi }{4}+x)$$$$f{}^{\prime }\left(x\right)=0$$$$\Rightarrow cos(\frac{\pi }{4}+x)=0$$$$\Rightarrow x=2n\pi \pm \frac{\pi }{2}-\frac{\pi }{4}=2n\pi +\frac{\pi }{4},2n\pi -\frac{3\pi }{4}$$$$whichdivides[0,2\pi ]into3intervals$$$$[0,\frac{\pi }{4}),(\frac{\pi }{4},\frac{5\pi }{4}),(\frac{5\pi }{4},2\pi ]$$$$\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{r}\mathit{v}\mathit{a}\mathit{l}\mid \mathit{s}\mathit{i}\mathit{g}\mathit{n}\mathit{o}\mathit{f}\mathit{f}{}^{\prime }\left(\mathit{x}\right)\mid \mathit{n}\mathit{a}\mathit{t}\mathit{u}\mathit{r}\mathit{e}\mathit{o}\mathit{f}\mathit{f}\left(\mathit{x}\right)$$$$[0,\frac{\pi }{4})\mid (+)\mid strictlyincreasing$$$$(\frac{\pi }{4},\frac{5\pi }{4})\mid (-)\mid strictlydecreasing$$$$(\frac{5\pi }{4},2\pi ]\mid (+)\mid stictlyincreasing$$$$\therefore f\left(x\right)isincreasingin[0,\frac{\pi }{4})\cup (\frac{5\pi }{4},2\pi ]$$$$f\left(x\right)isdecreasingin(\frac{\pi }{4},\frac{5\pi }{4})$$$$[\mathit{N}\mathit{o}\mathit{t}\mathit{e}:thisanswerisnotvalidforevery$$$$valueofx,onlyforthedomain[0,2\pi ]]$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com