y = x + square root (absolute value (x))

what are the first and second derivatives?

and the extrema, concavity, any asymptotes, continuity?

I dont even know where to begin

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- Dec 22nd 2009, 12:40 PMdjdf92Analysis of graph
y = x + square root (absolute value (x))

what are the first and second derivatives?

and the extrema, concavity, any asymptotes, continuity?

I dont even know where to begin - Dec 23rd 2009, 12:53 AMKiwi_Dave
The absolute value causes us some problems. To overcome this you should probably analyse the problem for the three seperate cases of x>0, x<0 and x=0.

Case 1; For x>0

$\displaystyle y=x+\sqrt x$

$\displaystyle \therefore y'=1+ \frac {1}{2\sqrt x}$

$\displaystyle \therefore y''=\frac {-1}{4\sqrt{x^3}}$

Case 2; For x<0

$\displaystyle y=x+\sqrt {-x}$

$\displaystyle \therefore y'=1- \frac {1}{2\sqrt {-x}}$

$\displaystyle \therefore y''=\frac {-1}{4\sqrt{{-x}^3}}$

Using these equations you should be able to answer some of the questions above.