Question

If $a,b,c\ge 0: ab+bc+ca=1,$ prove $\frac{1}{\sqrt{a+1}}+\frac{1}{\sqrt{b+1}}+\frac{1}{\sqrt{c+1}}\le 1+\sqrt{2}.$

score 225 · Answer 1 · 2023-08-01 08:35:23

225

Views

If $a,b,c\ge 0: ab+bc+ca=1,$ prove $\frac{1}{\sqrt{a+1}}+\frac{1}{\sqrt{b+1}}+\frac{1}{\sqrt{c+1}}\le 1+\sqrt{2}.$

Published on 01 Aug 2023 - 8:35

score 380 · Answer 2 · 2023-08-01 16:32:24

380

Views

If $x,y,z\ge 1: xy+yz+zx=2(x+y+z-1),$ find the smallest constant $K$ such that $\sqrt{x}+\sqrt{y}+\sqrt{z}\le K\sqrt{xyz}.$

Published on 01 Aug 2023 - 16:32

score 156 · Answer 3 · 2023-08-02 06:19:35

156

Views

If $xy+yz+zx+xyz=4,$ prove $\frac{1}{\sqrt{x+1}}+\frac{1}{\sqrt{y+1}}+\frac{1}{\sqrt{z+1}} \leq 1+\frac{2}{\sqrt{3}}.$

Published on 02 Aug 2023 - 6:19

score 15 · Answer 4 · 2023-08-06 12:11:45

15

Views

Gradient of an integration functional

Published on 06 Aug 2023 - 12:11

score 80 · Answer 5 · 2026-03-26 06:32:36

80

Views

Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?

Published on 26 Mar 2026 - 6:32

score 192 · Answer 6 · 2026-03-26 21:09:14

192

Views

How to use the Karush–Kuhn–Tucker conditions properly?

Published on 26 Mar 2026 - 21:09

score 142 · Answer 7 · 2026-03-25 14:36:44

142

Views

Finding $\small{\max\limits_{a+b+c=3}\frac{1}{\sqrt{a^2-3a+3}}+\frac{1}{\sqrt{b^2-3b+3}}+\frac{1}{\sqrt{c^2-3c+3}}.}$

Published on 25 Mar 2026 - 14:36

score 28 · Answer 8 · 2023-08-09 15:16:47

28

Views

Problem on Lagrange multiplier (gradient method )

Published on 09 Aug 2023 - 15:16

score 111 · Answer 9 · 2026-03-27 14:28:37

111

Views

Derivation of closed form lasso dual

Published on 27 Mar 2026 - 14:28

score 183 · Answer 10 · 2023-08-11 15:53:19

183

Views

Why the Lagrangian function is has different sign depending on the book?

Published on 11 Aug 2023 - 15:53

score 269 · Answer 11 · 2023-08-15 08:21:33

269

Views

Lagrange Multipliers Question - some extremum points are missed (not detected) by the method

Published on 15 Aug 2023 - 8:21

score 85 · Answer 12 · 2026-02-23 17:23:08

85

Views

Turning a constrained optimal control problem into unconstrained (Lagrangian)

Published on 23 Feb 2026 - 17:23

score 48 · Answer 13 · 2026-03-26 21:13:33

48

Views

Geometric Proof About Constrained Minimization Problem

Published on 26 Mar 2026 - 21:13

score 200 · Answer 14 · 2023-08-16 11:17:06

200

Views

Cobb-Douglas model - Lagrange multipliers

Published on 16 Aug 2023 - 11:17

score 74 · Answer 15 · 2023-08-17 17:29:08

74

Views

Show that Lagrange multiplier method fails to solve $\min(x^2+y^2)$ subject to $(x-1)^3-y^2=0$

Published on 17 Aug 2023 - 17:29

List Question

If $a,b,c\ge 0: ab+bc+ca=1,$ prove $\frac{1}{\sqrt{a+1}}+\frac{1}{\sqrt{b+1}}+\frac{1}{\sqrt{c+1}}\le 1+\sqrt{2}.$

If $x,y,z\ge 1: xy+yz+zx=2(x+y+z-1),$ find the smallest constant $K$ such that $\sqrt{x}+\sqrt{y}+\sqrt{z}\le K\sqrt{xyz}.$

If $xy+yz+zx+xyz=4,$ prove $\frac{1}{\sqrt{x+1}}+\frac{1}{\sqrt{y+1}}+\frac{1}{\sqrt{z+1}} \leq 1+\frac{2}{\sqrt{3}}.$

Gradient of an integration functional

Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?

How to use the Karush–Kuhn–Tucker conditions properly?

Finding $\small{\max\limits_{a+b+c=3}\frac{1}{\sqrt{a^2-3a+3}}+\frac{1}{\sqrt{b^2-3b+3}}+\frac{1}{\sqrt{c^2-3c+3}}.}$

Problem on Lagrange multiplier (gradient method )

Derivation of closed form lasso dual

Why the Lagrangian function is has different sign depending on the book?

Lagrange Multipliers Question - some extremum points are missed (not detected) by the method

Turning a constrained optimal control problem into unconstrained (Lagrangian)

Geometric Proof About Constrained Minimization Problem

Cobb-Douglas model - Lagrange multipliers

Show that Lagrange multiplier method fails to solve $\min(x^2+y^2)$ subject to $(x-1)^3-y^2=0$

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