your questions aren't specific enough, especially the "huh?" one
ok, let's try to get at the concept here. what does it mean for f' to be positive or negative? if the derivative is positive, it means our function is increasing, that is, going from bottom left to top right. if it is negative, it is decreasing, going from top left to bottom right. so we are saying, there is a point c where the derivative is zero. we want to find out what kind of critical point this is (that is, local max/min or inflection point). we simply test the sign of the derivative on both sides of the point. obviously if we are going up on the left and down on the right, we have a hill, which makes it a maximum. if we are going down on the left and up on the right, we have a valley, which is a minimum.1. If f' changes sign from positive to negative at c *f' > 0 for x , c and f' < 0 for x. c), then f has a local maximum value.\
The whole derivative or the y f' value from the derivative after plugging in c? Or the derivative equation (started as derived as [I am aware that is impossible, just bear with me for example sake]
thus, if f' > 0 to the left of c, and f' < 0 to the right of c, we have a local maximum
in the words of the wise Truthbetold, "Same issue."
2.If f' changes sign from negative to positive at c (f' < 0 for x < c and f' > 0 for x > c,) then f has a local maximum at c.
again, "Same issue." in this case, we have neither a hill or valley. but a possible inflection point
3. If f' does not change sign at c (f' has the same sing of both sides of c), then f has no local extreme value at c.
huh? what does "f' < 0 (f' > 0)" mean? i suppose you copied something wrong here, or you're using some notation peculiar to your class or textAt a left endpoint a:
If f' < 0 (f' > 0) for x > a, then f has a local maximum (minimum) value at a.
At a right endpoint a:
If f' < 0 (f'> 0) for x < b, then f has a local minimum (maximum) value a b.
EDIT: Oh, i get it, the brackets mean "respectively" in that case, "same issue" provided f' = 0 at the end point we are talking about. otherwise, it's hard to say it is a local max. there are other things to consider