Question

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

Answer 1

horizontal and vertical asymptotes

Answer 2

nope

Answer 3

@zepdrix

Answer 4

you're thinking of a saddle point maybe

http://en.wikipedia.org/wiki/Saddle_point

Answer 5

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

Answer 6

true, that's another name for it

saddle points apply to 1 variable functions as well

Answer 7

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

Answer 8

yes

Answer 9

something like this

Answer 10

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

Answer 11

well here's another example of a sharp point

Answer 12

aka, the absolute value function

Answer 13

hmmm... interesting.

Answer 14

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

Answer 15

well that's part of the definition anyway

Answer 16

So why does it show up as a critical value?

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

not sure what you mean

Answer 21

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?

Answer 22

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

Answer 23

This produces a local minimum at (0,0), so in this case, the cusp produced a minimum.