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horizontal and vertical asymptotes
you're thinking of a saddle point maybe http://en.wikipedia.org/wiki/Saddle_point
the answer is a sharp corner or an inflection point. Such a general question, I think it's just for single variable
true, that's another name for it saddle points apply to 1 variable functions as well
What is a "sharp corner" is that like a cusp?
something like this |dw:1353381760893:dw|
same same? Could it not be a vertical asymptote as well? Or what about a single point hole?
well here's another example of a sharp point |dw:1353381837898:dw|
aka, the absolute value function
a sharp point is basically a point that is non-differentiable, but the function is still continuous
well that's part of the definition anyway
So why does it show up as a critical value?
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
so if you look at the derivative alone, you'll see more critical points than extrema
If a cusp is non-differentiable are we able to still put a tangent line there?
not sure what you mean
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?
true, now I'm not sure, maybe there are some other criteria for a point to be considered a critical point
|dw:1353382502904:dw||dw:1353382579141:dw|This produces a local minimum at (0,0), so in this case, the cusp produced a minimum.
so maybe a cusp/sharp point isn't a point in which is a critical point, but not an extrema
Who knows, I'm going to try and find some solid definitions.
thanks for the input.