- The maximum of $x$ and $y$
- The minimum of $x$ and $y$
- $|x+y|$
- The average of $|x|$ and $|y|$
- The average of $|x+y|$ and $x-y$

Solution :

For all real numbers $x$ and $y$, which of the answers below is the expression \[\frac{x+y+|x-y|}{2}\] equal to?

- The maximum of $x$ and $y$
- The minimum of $x$ and $y$
- $|x+y|$
- The average of $|x|$ and $|y|$
- The average of $|x+y|$ and $x-y$

Solution :

The expression is equal to the maximum of $x$ and $y$.

Suppose $x>y$. Then $|x-y|=x-y$ and so \[\frac{x+y+|x-y|}{2}=\frac{x+y+x-y}{2}=x.\]

Now suppose $x<y$. Then $|x-y|=-(x-y)$ and so \[\frac{x+y+|x-y|}{2}=\frac{x+y-(x-y)}{2}=y.\]

\[\therefore\quad\frac{x+y+|x-y|}{2}=\max\{x,y\}.\]

Can you find a representation for $\min\{x,y\}$?

Suppose $x>y$. Then $|x-y|=x-y$ and so \[\frac{x+y+|x-y|}{2}=\frac{x+y+x-y}{2}=x.\]

Now suppose $x<y$. Then $|x-y|=-(x-y)$ and so \[\frac{x+y+|x-y|}{2}=\frac{x+y-(x-y)}{2}=y.\]

\[\therefore\quad\frac{x+y+|x-y|}{2}=\max\{x,y\}.\]

Can you find a representation for $\min\{x,y\}$?

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This webpage is LaTeX enabled. To type in-line formulae, type your stuff between two '$'. To type centred formulae, type '\[' at the beginning of your formula and '\]' at the end.

For the minimum of x and y, one would replace the expresion with $\frac{x+y-|x-y|}{2}$ (hopefully the tex works!)

Yup! You are correct! And the TeX works :)

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