function behavior

• Aug 9th 2009, 05:02 PM
live_laugh_luv27
function behavior
Which of the following describes the behavior of the graph
$\displaystyle y=\sqrt[4]{|x-2|}$ at $\displaystyle x=2$?

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

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

Originally Posted by live_laugh_luv27
Which of the following describes the behavior of the graph
$\displaystyle y=\sqrt[4]{|x-2|}$ at $\displaystyle x=2$?

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.
• Aug 9th 2009, 08:06 PM
CaptainBlack
Quote:

Originally Posted by live_laugh_luv27
Which of the following describes the behavior of the graph
$\displaystyle y=\sqrt[4]{|x-2|}$ at $\displaystyle x=2$?

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

CB
• Aug 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 $\displaystyle \rightarrow - \infty$ and right hand limit $\displaystyle \rightarrow + \infty$.