What two graphical features may occur at a critical value that does not generate a maximum or a minimum?

- baldymcgee6

- katieb

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- anonymous

horizontal and vertical asymptotes

- baldymcgee6

nope

- baldymcgee6

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## More answers

- jim_thompson5910

you're thinking of a saddle point maybe
http://en.wikipedia.org/wiki/Saddle_point

- baldymcgee6

the answer is a sharp corner or an inflection point. Such a general question, I think it's just for single variable

- jim_thompson5910

true, that's another name for it
saddle points apply to 1 variable functions as well

- baldymcgee6

What is a "sharp corner" is that like a cusp?

- jim_thompson5910

yes

- jim_thompson5910

something like this
|dw:1353381760893:dw|

- baldymcgee6

same same? Could it not be a vertical asymptote as well? Or what about a single point hole?

- jim_thompson5910

well here's another example of a sharp point
|dw:1353381837898:dw|

- jim_thompson5910

aka, the absolute value function

- baldymcgee6

hmmm... interesting.

- jim_thompson5910

a sharp point is basically a point that is non-differentiable, but the function is still continuous

- jim_thompson5910

well that's part of the definition anyway

- baldymcgee6

So why does it show up as a critical value?

- jim_thompson5910

well if there's a horizontal tangent at this sharp point or saddle point, then the derivative function will have a zero at that point

- jim_thompson5910

so if you look at the derivative alone, you'll see more critical points than extrema

- baldymcgee6

If a cusp is non-differentiable are we able to still put a tangent line there?

- jim_thompson5910

not sure what you mean

- baldymcgee6

You said at the sharp point it is continuous but not differentiable. I thought we had to be able to differentiate it to have a slope of 0 there?

- jim_thompson5910

true, now I'm not sure, maybe there are some other criteria for a point to be considered a critical point

- baldymcgee6

|dw:1353382502904:dw||dw:1353382579141:dw|This produces a local minimum at (0,0), so in this case, the cusp produced a minimum.

- jim_thompson5910

so maybe a cusp/sharp point isn't a point in which is a critical point, but not an extrema

- baldymcgee6

Who knows, I'm going to try and find some solid definitions.

- jim_thompson5910

alright

- baldymcgee6

thanks for the input.

- jim_thompson5910

np

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