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

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Presentation transcript:

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

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.

  is increasing  f is concave up   is decreasing  f is concave down changes signs  has a local extrema  f has an inflection point

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.

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?

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?

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