Section 2.6 Inflection Points and the Second Derivative Note: This is two day lecture, marked by and Calculator Required on all Sample Problems

f “ (a) gives us information on the concavity of the graph. If, the graph is concave up Slopes are positive Coordinates are positive

f “ (a) gives us information on the concavity of the graph. If, the graph is concave down Slopes are negative Coordinates are negative

If f” (a) = 0, a point of inflection MAY exist A point of inflection occurs when concavity changes Pt. of Inflection NO Pt. of Inflection

A.Find the x-coordinates of the point(s) of inflection B.On what interval(s) is f “ (x) < 0? C.On what interval(s) is f ‘ (x) > 0? (-1, 1), (3, 5) D.Where is a relative minimum of f(x)? x = 3 This is the graph of f(x) on (-1, 5) 2 (-1, 2)

A.Find the x-coordinates of the point(s) of inflection B.On what interval(s) is f “ (x) < 0? C.On what interval(s) is f ‘ (x) > 0? (0, 5) D.Where is a relative minimum of f(x)? x = 0 This is the graph of on (-1, 5) 1, 3 (1, 3)

A.Find the x-coordinates of the point(s) of inflection B.On what interval(s) is f “ (x) < 0? C.On what interval(s) is f ‘‘ (x) > 0? (0, 5) This is the graph of on (-1, 5) 0 (-1, 0)

A. Where is/are the point(s) of inflection? B.On what interval(s) is f ‘ (x) increasing? C. On what interval(s) is f “ (x) < 0? D.On what interval(s) do f “ (x) and f ‘ (x) have opposite signs? This is the graph of f(x) on [-2, 2]. -1.5, -0.5, 0.5, 1.5 (-1.5, -0.5), (0.5, 1.5) (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-1.5, -1), (-0.5, 0) (0.5, 1), (1.5, 2)

A. Where is/are the point(s) of inflection? B.On what interval(s) is f(x) increasing? C. On what interval(s) is f “ (x) < 0? D.On what interval(s) do f “ (x) and f ‘ (x) have opposite signs? This is the graph of on [-2, 2]. -1, 0, 1 (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-2, -1), (0, 1) (-2, -1.5), (-1, -0.5) (0, 0.5), (1, 1.5)

A. Where is/are the point(s) of inflection? B.On what interval(s) is f(x) concave up? C. On what interval(s) is f “ (x) < 0? This is the graph of on [-2, 2]. -1.5, -0.5, 0.5, 1.5 (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-1.5, -0.5), (0.5, 1.5)

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

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

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

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

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

A. On what interval(s) is f(x) concave up? B.Find the x-coordinate of the point(s) of inflection. C.On what interval(s) is f “ (x) > 0? This is the graph of f “ (x) on [-1, 5]. [-1, 1), (3, 5] 1, 3 [-1, 1), (3, 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]? This is the graph of f ‘ (x) on [-10, 3]. none (-10, 0), (0, 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 A) I only B) II only C) III only D) II and III only E) I, II, and III increasing NO TRUE

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 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 A)I only B) I and II only C) I and III only D) II and III only E) I, II and III NO Use a = 2