Unit 5: Analytical Applications of Differentiation

If a function [math] has a horizontal tangent at [math] and [math] while [math], what can be concluded about [math]?

The function [math] has a local maximum at

On [math], the number of inflection points of [math] is

The absolute maximum value of [math] on [math] is

If [math] for all [math] in [math] and [math] is differentiable on [math], which must be true? I. [math] is increasing on [math]. II. [math] is concave up on [math]. III. [math] has no inflection points on [math].

The Mean Value Theorem guarantees that for [math] on [math], there exists [math] such that [math]

If [math] and [math] for all [math] in an interval, then [math] is

[math]. Find all local extrema.

Find the absolute maximum of [math] on [math].

Where does [math] have an inflection point?

Use the Mean Value Theorem: if [math] on [math], find [math] such that [math].

On what interval is [math] increasing?

[math]. Find the intervals where [math] is concave up.

If [math] has a critical point at [math] and [math], what can be concluded?

The function [math] is continuous on [math]. [math]. At which value of [math] does [math] have a local minimum?

Let [math] be a twice-differentiable function. The table shows values of [math] at selected values of [math]. [math]: 0, 1, 2, 3, 4 [math]: [math], [math], [math], [math], [math] \nBased on this data, on which interval must [math] have a point of inflection?