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(Not) The First Derivative Test

Not the First Derivative Test

The "Not the First Derivative Test" is very simple, and it's intuitive, this just puts it into words.

If f'(x) > 0 for all x in (a,b), then f is increasing on [a,b]

If f'(x) < 0 for all x in (a,b), then f is decreasing on [a,b]

If f'(x) = 0 for all x in (a,b), then f is constant on [a,b]

The First Derivative Test

The First Derivative Test says that given a critical point, where the derivative is 0 or und.,

If f'(x) changes from negative to positive at c, then f is a relative minimum at (c, f(c)).

If f'(x) changes from positive to negative at c, then f is a relative maximum at (c, f(c)).

If f'(x) is positive or negative on both sides of c, then it doesn't mean anything.

 

David Witten

Differentials

Rolle's Theorem and the Mean Value Theorem