function behavior

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August 9th 2009, 05:02 PM
live_laugh_luv27

function behavior

Which of the following describes the behavior of the graph
$y=\sqrt[4]{|x-2|}$ at $x=2$ ?

1. differentiable
2. corner
3. cusp
4. vertical tangent
5. discontinuity

I think its a cusp..but not sure.
Thanks!
August 9th 2009, 07:54 PM
mr fantastic

Quote:

Originally Posted by live_laugh_luv27

Which of the following describes the behavior of the graph
$y=\sqrt[4]{|x-2|}$ at $x=2$ ?

1. differentiable
2. corner
3. cusp
4. vertical tangent
5. discontinuity

I think its a cusp..but not sure.
Thanks!

Cusp.

Since the tangent does not exist at x = 2 but the curve is continuous, only options 2 and 3 are candidates. But if you look at the definition of each, the answer has to be cusp.
August 9th 2009, 08:06 PM
CaptainBlack

Quote:

Originally Posted by live_laugh_luv27

Which of the following describes the behavior of the graph
$y=\sqrt[4]{|x-2|}$ at $x=2$ ?

1. differentiable
2. corner
3. cusp
4. vertical tangent
5. discontinuity

I think its a cusp..but not sure.
Thanks!

The left and right derivatives go to $\mp$ infinity as $x$ goes to $2$ , which in the case of slopes are the same thing (the curve is vertical on both sides at $x=1$ ) which of course makes the point a cusp.

CB
August 10th 2009, 01:46 AM
mr fantastic

Quote:

Originally Posted by CaptainBlack

The left and right derivatives go to infinity as x goes to 2.

CB

Indeed. Left hand limit $\rightarrow - \infty$ and right hand limit $\rightarrow + \infty$ .

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