Question

Given two positive numbers $x,\,y$ so that $32\,x^{6}+ 4\,y^{3}= 1$. Prove that $\frac{(2\,x^{2}+ y+ 3)^{5}}{3(x^{2}+ y^{2})- 3(x+ y)+ 2}\leqq 2048$ .

score 103 · Answer 1 · 2026-03-26 14:34:48

103

Views

Given two positive numbers $x,\,y$ so that $32\,x^{6}+ 4\,y^{3}= 1$. Prove that $\frac{(2\,x^{2}+ y+ 3)^{5}}{3(x^{2}+ y^{2})- 3(x+ y)+ 2}\leqq 2048$ .

Published on 26 Mar 2026 - 14:34

score 60 · Answer 2 · 2019-06-27 01:28:36

60

Views

Find the derivative of a multivariable Integral

Published on 27 Jun 2019 - 1:28

score 43 · Answer 3 · 2019-06-27 06:21:29

43

Views

Minimum and Maximum range of $\frac{\cot A+\cot B}{\sin^2C} + \frac{2\cot B+2\cot C}{\sin^2 A} - \frac{3\cot C+3\cot A}{\sin^2 B} $

Published on 27 Jun 2019 - 6:21

score 193 · Answer 4 · 2026-03-25 07:45:51

193

Views

Given three positive numbers $a,b,c$ so that $abc= 1$. Prove $(a-1+\frac{1}{b})(b-1+\frac{1}{c})(c-1+\frac{1}{a})\leqq\frac{2}{a+b+c-1}$ .

Published on 25 Mar 2026 - 7:45

score 209 · Answer 5 · 2026-03-25 07:44:59

209

Views

Calculate the maximum value of $|ab(a^2 - b^2) + bc(b^2 - c^2) + ca(c^2 - a^2)|$ where $a^2 + b^2 + c^2 = 1$.

Published on 25 Mar 2026 - 7:44

score 44 · Answer 6 · 2026-03-26 01:28:58

44

Views

Change of variables/substitution

Published on 26 Mar 2026 - 1:28

score 221 · Answer 7 · 2019-06-30 10:01:57

221

Views

Integration by Substitution in $\mathbb{R}^{n}$

Published on 30 Jun 2019 - 10:01

score 327 · Answer 8 · 2026-03-25 07:46:21

327

Views

Given three positive numbers $x,y,z$, prove that $(xyz+x^{2}y+y^{2}z+z^{2}x)^{4}\geqq\frac{256}{27}(x+y+z)^{3}x^{3}y^{3}z^{3}$ .

Published on 25 Mar 2026 - 7:46

score 145 · Answer 9 · 2019-07-01 10:16:41

145

Views

Rate of price depreciation

Published on 01 Jul 2019 - 10:16

score 67 · Answer 10 · 2019-07-01 19:16:39

67

Views

Integration of a natural logarithmic function using substitution

Published on 01 Jul 2019 - 19:16

score 65 · Answer 11 · 2019-07-01 19:29:32

65

Views

Change of variable by substitution

Published on 01 Jul 2019 - 19:29

score 215 · Answer 12 · 2026-03-25 23:39:40

215

Views

Solve integral $\int_0^\infty \frac{1}{\sqrt{x^2+1}(x^2+x+1)}dx$ using substitution $x=\sinh t$

Published on 25 Mar 2026 - 23:39

score 192 · Answer 13 · 2019-07-03 09:56:15

192

Views

Prove that $f(x)=x$.

Published on 03 Jul 2019 - 9:56

score 49 · Answer 14 · 2019-07-03 13:42:59

49

Views

how does this integration by substitution in Bernoulli's brachistochrone problem work?

Published on 03 Jul 2019 - 13:42

score 179 · Answer 15 · 2026-03-25 07:45:37

179

Views

Solve for $x,y,z \in \mathbb{R^+}$ ,$x^2+y^2+z^2=xyz+4$ and $xy+yz+zx=2(x+y+z)$

Published on 25 Mar 2026 - 7:45

List Question

Given two positive numbers $x,\,y$ so that $32\,x^{6}+ 4\,y^{3}= 1$. Prove that $\frac{(2\,x^{2}+ y+ 3)^{5}}{3(x^{2}+ y^{2})- 3(x+ y)+ 2}\leqq 2048$ .

Find the derivative of a multivariable Integral

Minimum and Maximum range of $\frac{\cot A+\cot B}{\sin^2C} + \frac{2\cot B+2\cot C}{\sin^2 A} - \frac{3\cot C+3\cot A}{\sin^2 B} $

Given three positive numbers $a,b,c$ so that $abc= 1$. Prove $(a-1+\frac{1}{b})(b-1+\frac{1}{c})(c-1+\frac{1}{a})\leqq\frac{2}{a+b+c-1}$ .

Calculate the maximum value of $|ab(a^2 - b^2) + bc(b^2 - c^2) + ca(c^2 - a^2)|$ where $a^2 + b^2 + c^2 = 1$.

Change of variables/substitution

Integration by Substitution in $\mathbb{R}^{n}$

Given three positive numbers $x,y,z$, prove that $(xyz+x^{2}y+y^{2}z+z^{2}x)^{4}\geqq\frac{256}{27}(x+y+z)^{3}x^{3}y^{3}z^{3}$ .

Rate of price depreciation

Integration of a natural logarithmic function using substitution

Change of variable by substitution

Solve integral $\int_0^\infty \frac{1}{\sqrt{x^2+1}(x^2+x+1)}dx$ using substitution $x=\sinh t$

Prove that $f(x)=x$.

how does this integration by substitution in Bernoulli's brachistochrone problem work?

Solve for $x,y,z \in \mathbb{R^+}$ ,$x^2+y^2+z^2=xyz+4$ and $xy+yz+zx=2(x+y+z)$

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