Ok, I had a function and the directions were:
Locate and classify all extrema as absolute or relative...
I can't remember the equation, but when i found the critical numbers of the function.. i got a critical number at the origin.. but when i graphed it on the calculator.. i got the graph above.. what is (0,0).. Is it anything? I don't think it's a max or min at all.. why would i get a crit # but no max or min.. just at an odd place like that... where the graph begins... To me, the answer would only be.. an absolute minimum at around (2, -2) or whatever it is (i didn't label the axes.. but you understand what i mean)
I can't be sure as you don't remember the function, but it is possibly because you didn't find the second derivative of the function. You see, the first derivative can equal zero indicating a local extrema. However, if the second derivative also equals zero at this point it is undefined. The case may be that the function did not have a real value at zero. This is what some call a discontinuity. There may even be a limit as the function approaches zero, but still have no value at that point. Again, the second derivative test would allow you to make this distinction.
Also, if the function had no values for x less than zero, it looks like there would be a relative maximum at zero.
why would i get a crit # but no max or min
It would be a saddle point. The slope of the graph is 0 at that point, but it is not a local max or min. For example, x^3 has a saddle point at (0,0).
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."