In this section, we will investigate some graphical relationships between a function and its second derivative.

Section 1.7 Geometry of Higher Order Derivatives

Definition

Let be the derivative of a function f. The second derivative of f, denoted is the derivative of .

So all of the relationships discussed in section 1.6 between a function and its derivative also exist between a function’s first and second derivatives.

Facts:

is increasing f is concave up

is decreasing f is concave down

changes signs has a local extrema

f has an inflection point

2nd Derivative Test

Suppose x = a is a stationary point of a function f.

That is, suppose . Then:

f has a local maximum at x = a.

f has a local minimum at x = a.

Example 1

Consider the graph of shown below.

(a) Where in [0, 5] is f concave down?

(b) Where in [0, 5] does have local minimums?

(c) Where in [0, 5] is increasing?

(d) Suppose . What type, if any, of extrema does f have at x = 3.5?

Example 2

Consider the graph of shown below.

(a) Where in [0, 5] is f increasing?

(b) Where in [0, 5] is ?

(c) Where does f have inflection points?

(d) Where does f have local maximums?

Example 3

Below are shown the graphs of a function as well as its first and second derivatives. Determine which each is.