Inflection Points and the Second Derivative Section 2.6 Day 2 Inflection Points and the Second Derivative I can analyze the graph of f ‘’(x) in order to draw conclusions about the graph of f (x). I can describe the graph of f ‘’(x) given f (x). Day 4 (Answer these on your bell ringer sheet) You are given the graph of f ’(x). For each of the graphs below, answer the following questions: What can you say about f(x)? What can you say about f”(x)? a. b.

A. For what value(s) of x is undefined?   On what interval(s) is f(x) concave down?. On what intervals is increasing? D. On what intervals is (-1, 1) (-3, -1), (1, 3) (-3, -1), (0, 1) -1, 1 This is the graph of f(x) on (-3, 3)

A. For what value(s) of x is undefined?   On what interval(s) is f(x) concave down?. On what intervals is increasing? D. On what intervals is (-3, -1), (0, 1) (-1, 0), (1, 3) none This is the graph of on (-3, 3)

On what interval(s) is f(x) concave up?   List the value(s) of x for which f(x) has a point of inflection. For what value(s) of x is ? none -1, 1 (-3, -1), (-1, 1), (1, 3) This is the graph of on (-3, 3)

A. For what value(s) of x is   On what intervals is f ‘ (x) > 0? C. On what intervals is f “ (x) < 0? Find the x-coordinate of the point(s) of inflection. -0.5, 0.5 (-2, -0.5), (0.5, 2) (-2, 0) x = 0 This is the graph of f(x) on (-2, 2)

A. For what value(s) of x is   On what intervals is f(x) decreasing? C. On what intervals is f “ (x) < 0? Find the x-coordinate of the point(s) of inflection. -1, 0, 1 (-2, -1), (0, 1) (-0.5, 0.5) -0.5, 0.5 This is the graph of f ‘ (x) on (-2, 2).

A. On what interval(s) is f(x) concave up?   Find the x-coordinate of the point(s) of inflection. On what interval(s) is f “ (x) > 0? [-1, 1), (3, 5] 1, 3 This is the graph of f “ (x) on [-1, 5].

For what value(s) of x does f ‘ (x) not exist?   On what interval(s) is f(x) concave down? On what interval(s) is f “ (x) > 0? Where is/are the relative minima on [-10, 3]? none (-10, 0), (0, 3) -1 This is the graph of f ‘ (x) on [-10, 3].

Which of the following is/are true about the function f if its derivative is defined by I) f is decreasing for all x < 4 II) f has a local maximum at x = 1 III) f is concave up for all 1 < x < 3 increasing NO TRUE A) I only B) II only C) III only D) II and III only E) I, II, and III

The graph of the second derivative of a function f is shown below. Which of the following are true about the original function f? I) The graph of f has an inflection point at x = -2 II) The graph of f has an inflection point at x = 3 III) The graph of f is concave down on the interval (0, 4) A) I only B) II only C) III only D) I and II only E) I, II and III NO YES NO

Which of the following statements are true about the function f, if it’s derivative f ‘ is defined by Use a = 2 I) The graph of f is increasing at x = 2a II) The function f has a local maximum at x = 0 III) The graph of f has an inflection point at x = a I only B) I and II only C) I and III only D) II and III only E) I, II and III NO