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Inequality that is Symmetrical but not Symmetric

1) Show that for any 3 real numbers \( a, b, c \), we have

\[ \sqrt{ a^2 - ab + b^2 } + \sqrt{ b^2 - bc + c^2 } \geq \sqrt{ a^2 + ac + c^2 }. \]

When does equality occur?


2) If \(a, b\) and \(c\) are real numbers such that \( a \geq c, b \geq c, c > 0 \), show that

\[ \sqrt{c(a-c) } + \sqrt{ c (b-c) } \leq \sqrt{ab} .\]

When does equality occur?


3) Let \(a, b\) and \(c\) be real numbers with absolute value less than or equal to 1. Show that

\[ ab+bc+ca + 1 \geq 0. \]

When does equality occur?


Hint: I used a tag of #Geometric Interpretation.

Note by Calvin Lin
3 years, 2 months ago

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Diagram for (2):

http://i.imgur.com/FXhnSs5.png

http://i.imgur.com/FXhnSs5.png

Jon Haussmann · 3 years, 2 months ago

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1) Angle between a and b is 60 deg, angle between b and c is 60 deg and it is a case of triangular inequality. Rajen Kapur · 3 years, 2 months ago

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