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Question Number 163210 Answers: 1 Comments: 1

Question Number 163209 Answers: 0 Comments: 0

f : I → (0 ; ∞) ; I ⊂ R f - twice derivable ; f^′ ; f^(′′) - continuous f^(′′) (x) f(x) ≥ (f^′ (x))^2 ; ∀ x ∈ I then prove that: 2f (((x + y)/2)) ≤ f(x) + f(y) ; ∀ x;y ∈ I

$$\mathrm{f}\::\:\mathrm{I}\:\rightarrow\:\left(\mathrm{0}\:;\:\infty\right)\:\:;\:\:\mathrm{I}\:\subset\:\mathbb{R} \\ $$$$\mathrm{f}\:-\:\mathrm{twice}\:\mathrm{derivable}\:\:;\:\:\mathrm{f}\:^{'} \:;\:\mathrm{f}\:^{''} \:-\:\mathrm{continuous} \\ $$$$\mathrm{f}\:^{''} \left(\mathrm{x}\right)\:\mathrm{f}\left(\mathrm{x}\right)\:\geqslant\:\left(\mathrm{f}\:^{'} \left(\mathrm{x}\right)\right)^{\mathrm{2}} \:;\:\:\forall\:\mathrm{x}\:\in\:\mathrm{I} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{2f}\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}}{\mathrm{2}}\right)\:\leqslant\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{y}\right)\:\:;\:\:\forall\:\mathrm{x};\mathrm{y}\:\in\:\mathrm{I} \\ $$

Question Number 163205 Answers: 0 Comments: 0

Fourier series expansion for ln(sin(x))

$$\boldsymbol{{F}}{ourier}\:{series}\:{expansion}\:{for}\:{ln}\left({sin}\left({x}\right)\right) \\ $$

Question Number 163197 Answers: 0 Comments: 1

Question Number 163191 Answers: 0 Comments: 1

Question Number 163175 Answers: 0 Comments: 1

Question Number 163171 Answers: 1 Comments: 0

Prove that sin 36° = ((√(10−2(√5^ )))/4)

$$\mathrm{Prove}\:\mathrm{that}\:\:\:\mathrm{sin}\:\mathrm{36}°\:=\:\frac{\sqrt{\mathrm{10}−\mathrm{2}\sqrt{\mathrm{5}^{} }}}{\mathrm{4}} \\ $$

Question Number 163214 Answers: 0 Comments: 1

lim_(x→0) ((2sin x−2tan x+x^3 )/(6x−2sin 3x−9x^3 )) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:{x}−\mathrm{2tan}\:{x}+{x}^{\mathrm{3}} }{\mathrm{6}{x}−\mathrm{2sin}\:\mathrm{3}{x}−\mathrm{9}{x}^{\mathrm{3}} }\:=? \\ $$

Question Number 163212 Answers: 1 Comments: 3

Question Number 163168 Answers: 2 Comments: 1

Question Number 163167 Answers: 1 Comments: 0

6^(x+1) +1 = 8^(x+1) −27^x x=?

$$\:\mathrm{6}^{{x}+\mathrm{1}} \:+\mathrm{1}\:=\:\mathrm{8}^{{x}+\mathrm{1}} −\mathrm{27}^{{x}} \: \\ $$$$\:{x}=? \\ $$

Question Number 163166 Answers: 0 Comments: 0

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Question Number 163163 Answers: 1 Comments: 1

x^9 −2022x^3 +(√(2021))=0 x={?}

$${x}^{\mathrm{9}} −\mathrm{2022}{x}^{\mathrm{3}} +\sqrt{\mathrm{2021}}=\mathrm{0} \\ $$$${x}=\left\{?\right\} \\ $$

Question Number 163161 Answers: 1 Comments: 0

Question Number 163158 Answers: 1 Comments: 0

prove that ∫_0 ^( (π/4)) (( sin(x)+cos(x))/( (√(1+sin(x)cos(x))))) dx= (√2) .cot^( −1) ((√2) ) −−−−−

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{\:{sin}\left({x}\right)+{cos}\left({x}\right)}{\:\sqrt{\mathrm{1}+{sin}\left({x}\right){cos}\left({x}\right)}}\:{dx}=\:\sqrt{\mathrm{2}}\:.{cot}^{\:−\mathrm{1}} \left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:−−−−− \\ $$

Question Number 163153 Answers: 2 Comments: 0

show that ((cos(x−y))/(cos(x+y)))=((1+tanxtany)/(1−tanxtany))

$${show}\:{that} \\ $$$$\:\frac{{cos}\left({x}−{y}\right)}{{cos}\left({x}+{y}\right)}=\frac{\mathrm{1}+{tanxtany}}{\mathrm{1}−{tanxtany}} \\ $$

Question Number 163148 Answers: 1 Comments: 4

Question Number 163144 Answers: 1 Comments: 0

Find: 𝛀 = ∫_( 0) ^( 1) (((π/4) - arctan(x))/(1 - x)) dx

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\frac{\pi}{\mathrm{4}}\:-\:\mathrm{arctan}\left(\mathrm{x}\right)}{\mathrm{1}\:-\:\mathrm{x}}\:\mathrm{dx} \\ $$

Question Number 163143 Answers: 0 Comments: 2

Question Number 163137 Answers: 1 Comments: 0

Soit U_n =((n−1)/(n^2 −5n+5)) avec n ∈ N. Montrer par la definition que U_n converge vers 0.

$${Soit}\:{U}_{{n}} =\frac{{n}−\mathrm{1}}{{n}^{\mathrm{2}} −\mathrm{5}{n}+\mathrm{5}}\:{avec}\:{n}\:\in\:\mathbb{N}. \\ $$$${Montrer}\:{par}\:{la}\:{definition}\:{que}\:{U}_{{n}} \: \\ $$$${converge}\:{vers}\:\mathrm{0}. \\ $$

Question Number 164451 Answers: 0 Comments: 1

How do you all to prove is true or false??; Prove the: (1/(1+p+p^2 +p^3 )) + (1/(1+q+q^2 +q^3 )) + (1/(1+r+r^2 +r^3 )) + (1/(1+s+s^2 +s^(3 ) )) ≥ 1

Question Number 163134 Answers: 1 Comments: 0

prove or disprove ∫_(2π) ^( 4π) (( sin(x))/x) dx >0 because ∫_(2π) ^( 3π) (( sin(x ))/x) dx > ∫_(3π) ^( 4π) ((∣sin(x)∣)/x) dx

$$ \\ $$$$\:\:\:\:{prove}\:\:{or}\:{disprove} \\ $$$$ \\ $$$$\:\:\:\:\int_{\mathrm{2}\pi} ^{\:\mathrm{4}\pi} \frac{\:{sin}\left({x}\right)}{{x}}\:{dx}\:>\mathrm{0} \\ $$$$\:\:\:\:\:\:\:{because} \\ $$$$\:\int_{\mathrm{2}\pi} ^{\:\mathrm{3}\pi} \frac{\:{sin}\left({x}\:\right)}{{x}}\:{dx}\:>\:\int_{\mathrm{3}\pi} ^{\:\mathrm{4}\pi} \frac{\mid{sin}\left({x}\right)\mid}{{x}}\:{dx} \\ $$$$ \\ $$

Question Number 163126 Answers: 0 Comments: 1

Please dear members. concerning this App How to change the color of the paper on which we are writing(for example from white to any kind of color)??????

$$\:\boldsymbol{{Please}}\:\boldsymbol{{dear}}\:\boldsymbol{{members}}.\:\boldsymbol{{concerning}}\:\boldsymbol{{this}}\:\boldsymbol{{App}} \\ $$$$\:\boldsymbol{{How}}\:\boldsymbol{{to}}\:\boldsymbol{{change}}\:\boldsymbol{{the}}\:\boldsymbol{{color}}\: \\ $$$$\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{paper}}\:\boldsymbol{{on}}\:\boldsymbol{{which}}\:\boldsymbol{{we}}\:\boldsymbol{{are}} \\ $$$$\:\boldsymbol{{writing}}\left(\boldsymbol{{for}}\:\boldsymbol{{example}}\:\boldsymbol{{from}}\:\boldsymbol{{white}}\:\boldsymbol{{to}}\:\boldsymbol{{any}}\:\boldsymbol{{kind}}\right. \\ $$$$\left.\:\:\boldsymbol{{of}}\:\boldsymbol{{color}}\right)?????? \\ $$

Question Number 163125 Answers: 0 Comments: 0

Question Number 163120 Answers: 1 Comments: 0

Question Number 163119 Answers: 2 Comments: 0

f ′(x)= f(x)+∫_0 ^1 f(x)dx f(0)=1 ⇒f(x)=?

$$\:\:{f}\:'\left({x}\right)=\:{f}\left({x}\right)+\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\: \\ $$$$\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:\Rightarrow{f}\left({x}\right)=? \\ $$

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