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Question Number 219846 Answers: 1 Comments: 0

If f:[a,b]→R 0<a≤b f - continuous Then prove that ((b^(4047) − a^(4047) )/(4047)) + ∫_a ^( b) f^2 (x^(2024) ) dx ≥ (1/(1012)) ∫_a^(2024) ^( b^(2024) ) f(x) dx

$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{f}\:-\:\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{b}^{\mathrm{4047}} \:−\:\mathrm{a}^{\mathrm{4047}} }{\mathrm{4047}}\:+\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\:^{\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{2024}} \right)\:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{1012}}\:\int_{\boldsymbol{\mathrm{a}}^{\mathrm{2024}} } ^{\:\boldsymbol{\mathrm{b}}^{\mathrm{2024}} } \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 219844 Answers: 2 Comments: 0

Question Number 219843 Answers: 0 Comments: 0

If a,b,c,d > 0 a^2 +b^2 +c^2 +d^2 = 4 Then prove that (1/((1+ab)^3 )) + (1/((1+ac)^3 )) + (1/((1+ad)^3 )) + (1/((1+bc)^3 )) + (1/((1+bd)^3 )) + (1/((1+cd)^3 )) ≥ (3/4)

$$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ab}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ac}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ad}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bc}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bd}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{cd}\right)^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 219839 Answers: 1 Comments: 0

Question Number 219833 Answers: 0 Comments: 0

Question Number 219832 Answers: 1 Comments: 0

lim_(n→∞) n((1/(1+n)) +(1/(2+n)) +...+(1/(2n)) −ln(2))=?

$$ \\ $$$$\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:{n}\left(\frac{\mathrm{1}}{\mathrm{1}+{n}}\:+\frac{\mathrm{1}}{\mathrm{2}+{n}}\:+...+\frac{\mathrm{1}}{\mathrm{2}{n}}\:−{ln}\left(\mathrm{2}\right)\right)=? \\ $$$$ \\ $$

Question Number 219831 Answers: 1 Comments: 0

prove lim_(h→0) (((g(z+h))/(g(z))))^(1/h) =e^(((d )/dz) ln (g(z))) =e^((g^((1)) (z))/(g(z)))

$$\mathrm{prove} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{g}\left({z}+{h}\right)}{\mathrm{g}\left({z}\right)}\right)^{\frac{\mathrm{1}}{{h}}} ={e}^{\frac{\mathrm{d}\:\:}{\mathrm{d}{z}}\:\mathrm{ln}\:\left(\mathrm{g}\left({z}\right)\right)} ={e}^{\frac{\mathrm{g}^{\left(\mathrm{1}\right)} \left({z}\right)}{\mathrm{g}\left({z}\right)}} \\ $$

Question Number 219806 Answers: 2 Comments: 0

lim_(h→0) (((cos(x+h))/(cos(x))))^(1/h) =??

$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left(\frac{\mathrm{cos}\left({x}+{h}\right)}{\mathrm{cos}\left({x}\right)}\right)^{\frac{\mathrm{1}}{{h}}} =?? \\ $$

Question Number 219801 Answers: 1 Comments: 0

lim_(h→∞) h^ν J_ν (h)=??

$$\underset{{h}\rightarrow\infty} {\mathrm{lim}}\:{h}^{\nu} {J}_{\nu} \left({h}\right)=?? \\ $$

Question Number 219800 Answers: 0 Comments: 0

y^((2)) (t)=(1−e^t )y(t)+y^((1)) (t)

$${y}^{\left(\mathrm{2}\right)} \left({t}\right)=\left(\mathrm{1}−{e}^{{t}} \right){y}\left({t}\right)+{y}^{\left(\mathrm{1}\right)} \left({t}\right) \\ $$

Question Number 219799 Answers: 0 Comments: 0

y^((2)) (t)+y(t)=cos(t)

$${y}^{\left(\mathrm{2}\right)} \left({t}\right)+{y}\left({t}\right)=\mathrm{cos}\left({t}\right) \\ $$

Question Number 219798 Answers: 0 Comments: 0

solve y^((2)) (t)=y^((1)) (t)e^(−y(t))

$$\mathrm{solve}\: \\ $$$${y}^{\left(\mathrm{2}\right)} \left({t}\right)={y}^{\left(\mathrm{1}\right)} \left({t}\right){e}^{−{y}\left({t}\right)} \\ $$

Question Number 219797 Answers: 0 Comments: 0

solve (y^((2)) (t))^2 =(1/(1+y(t)))

$$\mathrm{solve} \\ $$$$\left({y}^{\left(\mathrm{2}\right)} \left({t}\right)\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{1}+{y}\left({t}\right)} \\ $$

Question Number 219796 Answers: 0 Comments: 0

Solve y^((2)) (t)=(y(t))^2 −ay^((1)) (t)−by(t)

$$\mathrm{Solve} \\ $$$${y}^{\left(\mathrm{2}\right)} \left({t}\right)=\left({y}\left({t}\right)\right)^{\mathrm{2}} −{ay}^{\left(\mathrm{1}\right)} \left({t}\right)−{by}\left({t}\right) \\ $$

Question Number 219795 Answers: 1 Comments: 0

∫_(D=[0,1]^N ) Π_(h=1) ^N e^(−(1/2)x_h ) dx_h

$$\int_{\mathcal{D}=\left[\mathrm{0},\mathrm{1}\right]^{{N}} } \:\underset{{h}=\mathrm{1}} {\overset{{N}} {\prod}}\:{e}^{−\frac{\mathrm{1}}{\mathrm{2}}{x}_{{h}} } \mathrm{d}{x}_{{h}} \\ $$

Question Number 219794 Answers: 0 Comments: 0

∫_0 ^( ∞) ((Y_0 (t)e^(−3t) )/(t^2 +1)) dt

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{Y}_{\mathrm{0}} \left({t}\right){e}^{−\mathrm{3}{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t} \\ $$

Question Number 219793 Answers: 1 Comments: 0

∫_0 ^( ∞) (e^(−t) /(t^2 +1)) dt=?

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{e}^{−{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t}=? \\ $$

Question Number 219792 Answers: 1 Comments: 0

lim_(z→∞) ((J_1 (z))/(Y_0 (z))) lim_(z→0) z((1−(1/2)z^2 −cos((z/(1−z^2 ))))/z^4 ) lim_(z→0) ((J_ν (z+h)Y_ν (z)−J_ν (z)Y_ν (z))/h)

$$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{{J}_{\mathrm{1}} \left({z}\right)}{{Y}_{\mathrm{0}} \left({z}\right)} \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{z}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{z}^{\mathrm{2}} −\mathrm{cos}\left(\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\right)}{{z}^{\mathrm{4}} } \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{J}_{\nu} \left({z}+{h}\right){Y}_{\nu} \left({z}\right)−{J}_{\nu} \left({z}\right){Y}_{\nu} \left({z}\right)}{{h}}\: \\ $$

Question Number 219789 Answers: 1 Comments: 4

Question Number 219787 Answers: 2 Comments: 0

Question Number 219786 Answers: 1 Comments: 0

Question Number 219785 Answers: 3 Comments: 0

Question Number 219784 Answers: 8 Comments: 0

Question Number 219871 Answers: 1 Comments: 0

F(s)=∫_0 ^( ∞) ((sin(t))/(t^2 +α^2 ))e^(−st) dt F(3)=??

$${F}\left({s}\right)=\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{sin}\left({t}\right)}{{t}^{\mathrm{2}} +\alpha^{\mathrm{2}} }{e}^{−{st}} \:\mathrm{d}{t} \\ $$$$\mathrm{F}\left(\mathrm{3}\right)=?? \\ $$

Question Number 219770 Answers: 1 Comments: 3

Question Number 219769 Answers: 2 Comments: 1

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