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Category: Differentiation

prove-0-1-1-x-2-ln-3-1-x-x-dx-51-8-pi-4-15-m-n-

Question Number 166082 by mnjuly1970 last updated on 12/Feb/22 $$ \\ $$$$\:\:\:\:\:\:\:{prove} \\ $$$$\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\left(\mathrm{1}−{x}\:\right)^{\:\mathrm{2}} .{ln}^{\:\mathrm{3}} \left(\mathrm{1}−{x}\:\right)}{{x}}\:{dx}\:=\:\frac{\mathrm{51}}{\mathrm{8}}\:−\frac{\pi^{\:\mathrm{4}} }{\mathrm{15}}\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$\:\:\:\:\:\:\: \\ $$$$ \\ $$…

let-f-x-y-z-x-2-y-2-z-2-with-R-1-calculate-f-2-find-in-order-to-have-f-0-

Question Number 34912 by abdo imad last updated on 12/May/18 $${let}\:{f}\left({x},{y},{z}\right)\:=\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)^{\alpha} \:\:\:\:\:{with}\:\alpha\in{R} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\Delta{f} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\alpha\:{in}\:{order}\:{to}\:{have}\:\Delta{f}=\mathrm{0} \\ $$ Answered by tanmay.chaudhury50@gmail.com last…

Given-f-0-2-R-f-x-is-twice-derivable-and-f-0-f-1-f-2-0-i-Show-that-there-exist-c-1-c-2-such-that-f-c-1-0-and-f-c-2-0-ii-Show-that-there-exist-c-3-such-that-f-c-3-0-

Question Number 100388 by Ar Brandon last updated on 26/Jun/20 $$\:\:\:\:\:\:\:\mathcal{G}\mathrm{iven}\:\mathrm{f}:\left[\mathrm{0},\mathrm{2}\right]\rightarrow\mathbb{R}\:,\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{twice}\:\mathrm{derivable}\:\mathrm{and}\: \\ $$$$\mathrm{f}\left(\mathrm{0}\right)=\mathrm{f}\left(\mathrm{1}\right)=\mathrm{f}\left(\mathrm{2}\right)=\mathrm{0} \\ $$$${i}-\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{c}_{\mathrm{1}} ,\:\mathrm{c}_{\mathrm{2}} ,\:\mathrm{such}\:\mathrm{that}\:\mathrm{f}'\left(\mathrm{c}_{\mathrm{1}} \right)=\mathrm{0}\: \\ $$$$\mathrm{and}\:\mathrm{f}'\left(\mathrm{c}_{\mathrm{2}} \right)=\mathrm{0} \\ $$$${ii}-\mathcal{S}\mathrm{how}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{c}_{\mathrm{3}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{f}''\left(\mathrm{c}_{\mathrm{3}} \right)=\mathrm{0}…

Determine-the-coordinates-where-the-function-f-x-ax-2-bx-c-admits-a-local-point-

Question Number 100391 by Ar Brandon last updated on 26/Jun/20 $$\:\:\:\mathcal{D}\mathrm{etermine}\:\mathrm{the}\:\mathrm{coordinates}\:\mathrm{where}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c} \\ $$$$\mathrm{admits}\:\mathrm{a}\:\mathrm{local}\:\mathrm{point}. \\ $$ Answered by MJS last updated on 26/Jun/20 $${ax}^{\mathrm{2}} +{bx}+{c}={x}…

solve-the-differential-equation-dy-dx-y-x-1-1-x-1-

Question Number 165849 by daus last updated on 09/Feb/22 $${solve}\:{the}\:{differential}\:{equation} \\ $$$$\frac{{dy}}{{dx}}+\frac{{y}}{{x}−\mathrm{1}}=\frac{\mathrm{1}}{{x}+\mathrm{1}} \\ $$ Commented by mkam last updated on 10/Feb/22 $$\boldsymbol{{p}}\left(\boldsymbol{{x}}\right)=\:\frac{\mathrm{1}}{\boldsymbol{{x}}−\mathrm{1}}\:\:,\:\boldsymbol{{Q}}\left(\boldsymbol{{x}}\right)\:=\:\frac{\mathrm{1}}{\boldsymbol{{x}}+\mathrm{1}} \\ $$$$ \\…