I do not understand the following statements which describe the First Derivative Test for Local Extrema.

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 $\displaystyle y= x^2$ derived as $\displaystyle y'= - x^2$[I am aware that is impossible, just bear with me for example sake]

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.

Same issue.

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.

Same issue.

At a left endpoint a:

If f' < 0 (f' > 0) for x > a, then f has a local maximum (minimum) value at a.

Huh?

At a right endpoint a:

If f' < 0 (f'> 0) for x < b, then f has a local minimum (maximum) value a b.

Sane question.

Thank you!