EVT, Rolle's, MVT, and 1st Deriv Review

Front

What does it indicate when f'(x) changes from positive to negative?

Back

There is a relative maximum at that point.

Front

What does it indicate when f'(x) changes from negative to positive?

Back

There is a relative minimum at that point.

Front

If f'(x) < 0 over an interval, what can we say about f(x) in that interval?

Back

f(x) is decreasing over that interval.

Front

If f'(x) > 0 over an interval, what can we say about f(x) in that interval?

Back

f(x) is increasing over that interval.

Front

What is the significance of f'(a) = 0 at a point a?

Back

It indicates a potential relative extremum (maximum or minimum) at that point.

Front

What is Rolle's Theorem?

Back

If a function is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Front

What is the Mean Value Theorem (MVT)?

Back

If a function is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What does it mean if f'(x) does not exist (DNE) at a point?

Back

The function may have a corner, cusp, or vertical tangent at that point.

Front

What is the relationship between the first derivative and the increasing/decreasing behavior of a function?

Back

If f'(x) > 0, f(x) is increasing; if f'(x) < 0, f(x) is decreasing.

Front

What is a critical point?

Back

A point where f'(x) = 0 or f'(x) does not exist.