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Question Number 120342    Answers: 0   Comments: 0

Question Number 120331    Answers: 0   Comments: 8

Question Number 120325    Answers: 0   Comments: 6

Let f:R→R be a function satisfying the functional relation (f(x))^y +(f(y))^x =2f(xy) for all x, y ∈R and it is given that f(1)=1/2. Answer the following questions. (i) f(x+y)= (A) f(x)+f(y) (B) f(x)f(y) (C) f(x^y y^x ) (D) ((f(x))/(f(y))) (ii) f(xy)= (A) f(x)f(y) (B) f(x)+f(y) (C) (f(x))^y (D) (f(xy))^(xy) (iii) Σ_(k=0) ^∞ f(k)= (A) 5/2 (B) 3/2 (C) 3 (D) 2

$$\mathrm{Let}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{functional}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} +\left({f}\left(\mathrm{y}\right)\right)^{\mathrm{x}} =\mathrm{2}{f}\left(\mathrm{xy}\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{x},\:\mathrm{y}\:\in\mathbb{R}\:\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{given}\:\mathrm{that}\:{f}\left(\mathrm{1}\right)=\mathrm{1}/\mathrm{2}.\:\mathrm{Answer} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{questions}. \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\:\:\:\:{f}\left(\mathrm{x}+\mathrm{y}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:{f}\left(\mathrm{x}^{\mathrm{y}} \mathrm{y}^{\mathrm{x}} \right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\:\frac{{f}\left(\mathrm{x}\right)}{{f}\left(\mathrm{y}\right)} \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\:\:\:\:{f}\left(\mathrm{xy}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:{f}\left(\mathrm{x}\right){f}\left(\mathrm{y}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:{f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\left({f}\left(\mathrm{x}\right)\right)^{\mathrm{y}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\left({f}\left(\mathrm{xy}\right)\right)^{\mathrm{xy}} \\ $$$$\left(\boldsymbol{\mathrm{iii}}\right)\:\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({k}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{A}\right)\:\mathrm{5}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{3}/\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{2} \\ $$

Question Number 120324    Answers: 1   Comments: 0

we are in C. (E): z^3 +(4−5i)z^2 +(8−20i)z−40i=0 1) Show that (E) has one imaginary pure root 2) solve (E)

$$\mathrm{we}\:\mathrm{are}\:\mathrm{in}\:\mathbb{C}. \\ $$$$\left(\mathrm{E}\right):\:\mathrm{z}^{\mathrm{3}} +\left(\mathrm{4}−\mathrm{5i}\right)\mathrm{z}^{\mathrm{2}} +\left(\mathrm{8}−\mathrm{20i}\right)\mathrm{z}−\mathrm{40i}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{E}\right)\:\mathrm{has}\:\mathrm{one}\:\mathrm{imaginary}\:\mathrm{pure}\:\mathrm{root} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{solve}\:\left(\mathrm{E}\right) \\ $$

Question Number 120322    Answers: 1   Comments: 3

Question Number 120323    Answers: 0   Comments: 0

Question Number 120316    Answers: 1   Comments: 0

calculate ∫_0 ^(π/2) ((xcosx)/(cos(2x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{xcosx}}{{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 120312    Answers: 1   Comments: 0

... ♣nice calculus♣... prove that :: lim_(n→∞) (((n$))^(1/n^2 ) /( (√n)))?=^(???) e^((−3)/4) where :: n$ =^(superfactorial) n!.(n−1)!.(n−2)!...3!.2!.1! ...♠m.n.1970♠...

$$\:\:\:\:\:\:\:\:\:\:...\:\clubsuit{nice}\:\:{calculus}\clubsuit... \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \frac{\sqrt[{{n}^{\mathrm{2}} }]{{n\$}}}{\:\sqrt{{n}}}?\overset{???} {=}{e}^{\frac{−\mathrm{3}}{\mathrm{4}}} \\ $$$$\:\:\:\:\:{where}\:::\:\:{n\$}\:\overset{{superfactorial}} {=}{n}!.\left({n}−\mathrm{1}\right)!.\left({n}−\mathrm{2}\right)!...\mathrm{3}!.\mathrm{2}!.\mathrm{1}! \\ $$$$\:\:\:\:\:\:\:\:...\spadesuit{m}.{n}.\mathrm{1970}\spadesuit... \\ $$

Question Number 120526    Answers: 0   Comments: 1

Question Number 120301    Answers: 1   Comments: 3

is it possible to have(1+(1/x^4 ))^(1/2) =(1+(1/(2x^4 ))) please help

$${is}\:{it}\:{possible}\:{to}\:{have}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right)^{\frac{\mathrm{1}}{\mathrm{2}}} =\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{4}} }\right)\:{please}\:{help} \\ $$

Question Number 120300    Answers: 0   Comments: 1

Determine all function f:R→R which satisfy f(a+x)−f(a−x)=4ax for all real a and x.

$${Determine}\:{all}\:{function}\:{f}:{R}\rightarrow{R} \\ $$$${which}\:{satisfy}\:{f}\left({a}+{x}\right)−{f}\left({a}−{x}\right)=\mathrm{4}{ax} \\ $$$${for}\:{all}\:{real}\:{a}\:{and}\:{x}. \\ $$

Question Number 120297    Answers: 3   Comments: 0

∫_0 ^(π/2) ((x dx)/(sin x+cos x))

$$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{{x}\:{dx}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$

Question Number 120295    Answers: 1   Comments: 1

Question Number 120288    Answers: 1   Comments: 1

let f(x)=((ln(1+x^2 ))/(x^2 +3)) determine f^((n)) (x) and developp f at integr serie

$${let}\:{f}\left({x}\right)=\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} +\mathrm{3}} \\ $$$${determine}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$${and}\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 120285    Answers: 0   Comments: 0

calculate ∫_2 ^∞ (dx/((x^2 −1)^2 (x^2 +x+1)))

$${calculate}\:\:\int_{\mathrm{2}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)} \\ $$

Question Number 120284    Answers: 1   Comments: 1

Question Number 120283    Answers: 0   Comments: 0

fond ∫_2 ^∞ ((ln(1+3x^2 ))/(1+x^4 ))dx

$${fond}\:\int_{\mathrm{2}} ^{\infty} \frac{{ln}\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 120275    Answers: 1   Comments: 0

(((3+2 (5)^(1/4) )/(3−2 (5)^(1/4) )))^(1/(4 )) =?

$$\:\sqrt[{\mathrm{4}\:}]{\frac{\mathrm{3}+\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}{\mathrm{3}−\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}}\:=?\: \\ $$

Question Number 120274    Answers: 0   Comments: 0

Question Number 120272    Answers: 0   Comments: 0

ϕ(x)=x^r where x∈[0,+∞[ and 0<r≤1 shown that for any (x,y)∈R_+ ^2 𝛟(x+y)≤𝛟(x)+𝛟(y) please

$$\varphi\left({x}\right)=\boldsymbol{{x}}^{\boldsymbol{{r}}} \\ $$$$\boldsymbol{{where}}\:\boldsymbol{{x}}\in\left[\mathrm{0},+\infty\left[\:\boldsymbol{{and}}\:\mathrm{0}<\boldsymbol{{r}}\leqslant\mathrm{1}\right.\right. \\ $$$${shown}\:{that}\:{for}\:{any}\:\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)\in\mathbb{R}_{+} ^{\mathrm{2}} \\ $$$$\boldsymbol{\varphi}\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\leqslant\boldsymbol{\varphi}\left(\boldsymbol{{x}}\right)+\boldsymbol{\varphi}\left(\boldsymbol{{y}}\right) \\ $$$$\:\:\:\:\boldsymbol{{please}} \\ $$

Question Number 120268    Answers: 3   Comments: 0

Question Number 120258    Answers: 1   Comments: 0

Question Number 120257    Answers: 2   Comments: 1

∫ ((f ′(x))/(f(x))) =?

$$\:\int\:\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}\:=? \\ $$

Question Number 120254    Answers: 4   Comments: 0

∫ (dx/(1+cosθ.cos x )) ?

$$\:\int\:\frac{{dx}}{\mathrm{1}+\mathrm{cos}\theta.\mathrm{cos}\:{x}\:}\:? \\ $$

Question Number 120244    Answers: 3   Comments: 2

how to justify that sin (x−((7π)/2) )= cos x

$${how}\:{to}\:{justify}\:\:{that}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{7}\pi}{\mathrm{2}}\:\right)=\:\mathrm{cos}\:{x} \\ $$

Question Number 120243    Answers: 1   Comments: 0

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