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Question Number 142914 Answers: 0 Comments: 0

Let a≥b≥c≥d>0 and a+b+c+d = 4. Prove that (((√a)+(√b)+(√c))/3) ≤ (1/( (√d))) Prove if ∀n∈N^+ , then (((a)^(1/n) +(b)^(1/n) +(c)^(1/n) )/3) ≤ (1/( (d)^(1/n) ))

$$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant{d}>\mathrm{0}\:\mathrm{and}\:{a}+{b}+{c}+{d}\:=\:\mathrm{4}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt{{d}}} \\ $$$$\mathrm{Prove}\:\mathrm{if}\:\forall{n}\in\mathbb{N}^{+} ,\:\mathrm{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\sqrt[{{n}}]{{a}}+\sqrt[{{n}}]{{b}}+\sqrt[{{n}}]{{c}}}{\mathrm{3}}\:\leqslant\:\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{d}}} \\ $$$$ \\ $$

Question Number 142912 Answers: 0 Comments: 0

Question Number 142910 Answers: 2 Comments: 0

Question Number 142906 Answers: 1 Comments: 0

If abc=1 and a,b,c>0 prove that (a/(b^2 (c+1)))+(b/(c^2 (a+1)))+(c/(a^2 (b+1))) ≥ (3/2)

$${If}\:{abc}=\mathrm{1}\:{and}\:{a},{b},{c}>\mathrm{0}\:{prove} \\ $$$${that}\:\frac{{a}}{{b}^{\mathrm{2}} \left({c}+\mathrm{1}\right)}+\frac{{b}}{{c}^{\mathrm{2}} \left({a}+\mathrm{1}\right)}+\frac{{c}}{{a}^{\mathrm{2}} \left({b}+\mathrm{1}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 142904 Answers: 1 Comments: 0

If 2+log _2 (x)=3+log _3 (y)=log _6 (x−4y) then the value of (1/(2y))−(2/x)=?

$${If}\:\mathrm{2}+\mathrm{log}\:_{\mathrm{2}} \left({x}\right)=\mathrm{3}+\mathrm{log}\:_{\mathrm{3}} \left({y}\right)=\mathrm{log}\:_{\mathrm{6}} \left({x}−\mathrm{4}{y}\right) \\ $$$${then}\:{the}\:{value}\:{of}\:\frac{\mathrm{1}}{\mathrm{2}{y}}−\frac{\mathrm{2}}{{x}}=? \\ $$

Question Number 142902 Answers: 1 Comments: 0

I_n =∫_0 ^( _ (π/2)) (sin x)^n dx with integration by parts, prove that : I_(n+2) = ((n+1)/(n+2)) . I_n

$$\mathrm{I}_{{n}} =\int_{\mathrm{0}} ^{\:_{} \frac{\pi}{\mathrm{2}}} \:\left(\mathrm{sin}\:{x}\right)^{{n}} \:{dx} \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{integration}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{parts}},\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\: \\ $$$$\mathrm{I}_{{n}+\mathrm{2}} \:=\:\frac{{n}+\mathrm{1}}{{n}+\mathrm{2}}\:.\:\mathrm{I}_{{n}} \\ $$

Question Number 142901 Answers: 0 Comments: 2

Question Number 142900 Answers: 0 Comments: 3

Question Number 142898 Answers: 0 Comments: 0

Question Number 142903 Answers: 0 Comments: 0

Question Number 142893 Answers: 1 Comments: 0

.....mathematical .....analysis...... f ∈ C [0,1] and ∫_0 ^( 1) x^n f(x)dx=(1/(n+2)) , n∈N prove f(x):=x .....

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:.....{analysis}...... \\ $$$$\:\:\:\:\:\:\:{f}\:\in\:{C}\:\left[\mathrm{0},\mathrm{1}\right]\:{and}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{n}} {f}\left({x}\right){dx}=\frac{\mathrm{1}}{{n}+\mathrm{2}}\:,\:{n}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:{f}\left({x}\right):={x}\:..... \\ $$

Question Number 142886 Answers: 2 Comments: 0

Evaluate ∫_0 ^( (1/2)) ((4x^2 )/( (√(1−x^2 )) )) dx

$$\mathrm{Evaluate}\:\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{4}{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:}\:{dx} \\ $$

Question Number 142882 Answers: 2 Comments: 1

Question Number 142880 Answers: 1 Comments: 0

Prove that 𝛗(n)=nΠ_k (1−(1/p_k )) φ(n):Euler totient function

$$\:{Prove}\:{that}\:\boldsymbol{\phi}\left({n}\right)={n}\underset{{k}} {\prod}\left(\mathrm{1}−\frac{\mathrm{1}}{{p}_{{k}} }\right)\:\:\phi\left({n}\right):{Euler}\:{totient}\:{function} \\ $$

Question Number 142875 Answers: 1 Comments: 0

Prove that ζ(s)=Π_(prime) (1/(1−p^(−s) ))

$${Prove}\:{that}\:\zeta\left({s}\right)=\underset{{prime}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−{p}^{−{s}} } \\ $$

Question Number 142871 Answers: 0 Comments: 0

determine arctan(x+iy) at form u(x,y)+iv(x,y)

$$\mathrm{determine}\:\mathrm{arctan}\left(\mathrm{x}+\mathrm{iy}\right)\:\mathrm{at}\:\mathrm{form}\:\mathrm{u}\left(\mathrm{x},\mathrm{y}\right)+\mathrm{iv}\left(\mathrm{x},\mathrm{y}\right) \\ $$

Question Number 142870 Answers: 1 Comments: 0

Question Number 142869 Answers: 2 Comments: 0

calculate ∫_(−∞) ^(+∞) ((x^2 dx)/((x^2 −x+3)^2 ))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 142863 Answers: 1 Comments: 0

Question Number 142854 Answers: 0 Comments: 1

The maximum value of y=(√((x−3)^2 +(x^2 −2)^2 ))−(√(x^2 +(x−1)^2 )) is

$$\:\:{The}\:{maximum}\:{value}\:{of}\: \\ $$$$\:{y}=\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} }−\sqrt{{x}^{\mathrm{2}} +\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$$$\:{is}\: \\ $$

Question Number 142849 Answers: 1 Comments: 0

Question Number 142858 Answers: 1 Comments: 2

Question Number 142844 Answers: 2 Comments: 0

Given that f(sin x)=cos x, evaluate f ′(sin 45°).

$$\mathrm{Given}\:\mathrm{that}\:{f}\left(\mathrm{sin}\:{x}\right)=\mathrm{cos}\:{x},\:\mathrm{evaluate} \\ $$$${f}\:'\left(\mathrm{sin}\:\mathrm{45}°\right). \\ $$

Question Number 142829 Answers: 3 Comments: 0

Find the simplest form for T = (√(1+(√(−3)))) +(√(1−(√(−3))))

$$\:{Find}\:{the}\:{simplest}\:{form}\:{for}\: \\ $$$$\:\:{T}\:=\:\sqrt{\mathrm{1}+\sqrt{−\mathrm{3}}}\:+\sqrt{\mathrm{1}−\sqrt{−\mathrm{3}}}\: \\ $$

Question Number 142822 Answers: 1 Comments: 0

Question Number 142820 Answers: 0 Comments: 1

find k if : C_n ^k =n+1

$${find}\:{k}\:{if}\::\:{C}_{{n}} ^{{k}} ={n}+\mathrm{1} \\ $$

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