Problem of the Day (Calculator allowed) Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the following must be true? I. f has at least 2 zeroes II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f(c) = 3. A) None B) I only C) I and II only D) I and III only E) I, II, and III

Problem of the Day (Calculator allowed) Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the following must be true? I. f has at least 2 zeroes II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f(c) = 3. A) None B) I only C) I and II only D) I and III only E) I, II, and III I. 2 sign changes implies 2 zeroes II. Rolle's Theorem III. Intermediate Value Theorem

3-4: Concavity & The Second Derivative Test Objectives: Discuss concavity as an indicator of function behavior Recognize inflection as a change in the rate of change Use the 2nd Derivative Test ©2002 Roy L. Gover (www.mrgover.com)

Definition Concave up means the graph of f is above the tangent lines. f

Definition Concave down means the graph of f is below the tangent lines. f

Definition Concave up “holds water”

Definition Concave down “spills water”

Analysis Is there a relationship between the graphs of f(x) & f ’(x)? Is there a relationship between the concavity of f(x) and f’(x)? Is there a relationship between where concavity changes and f’(x)? Where does concavity change?

Definition f’(x) increasing Graph of f is concave up on interval I if f’ is increasing on I f’(x) increasing Concave Up

Definition f’(x) decreasing Graph of f is concave down on interval I if f’ is decreasing on I Concave Down f’(x) decreasing

f’ >0: slope of the tangent lines are positive; f is incr. Review f’ >0: slope of the tangent lines are positive; f is incr. f’ <0: slope of the tangent lines are negative; f is decr. f’ =0: slope of the tangent line is zero;f is neither increasing nor decreasing.

Important Idea f ’’>0:slope of tangent lines are becoming more positive (less negative) from left to right. f(x) f ‘(x) f”(x)

Important Ideas f(x) f ’’=0:slope of tangent lines are not changing. f ‘(x) f”(x)

Important Ideas f(x) f ’’<0:slope of tangent lines are becoming more negative (less positive) from left to right. f ‘(x) f”(x)

Important Idea Let f be a function such that f” exists on (a,b), then: f” (x)>0 for all x in (a,b) f is concave up. f” (x)<0 for all x in (a,b) f is concave down.

Procedure Determining Concavity: 1. Locate x values at which f ’’=0 or undefined. 2. Use these x values to determine intervals. 3. Test the sign of f ’’ in each interval

Example Determine the open intervals on which is concave up and concave down...

Example Step 1:Find the values of x where f ” =0 or undefined Step 2:Make a table using intervals determined in step 1 Step 3:Choose a value in each interval & evaluate the 2nd derivative at the value

Warm-Up Determine the open intervals on which the graph of is concave up and concave down.

Solution f(x) Up Down Interval Test Value -2 2 Sign f ”(x) + - Concl. 2 Sign f ”(x) + - Concl. Up Down

Definition Inflection point is the point where concavity changes. Inflection points occur where f’’(x)=0 or is undefined but f ”(x)=0 or undefined doesn’t guarantee an inflection point.

Important Idea An inflection point is the point where concavity changes. An inflection point is where the rate of change changes from increasing to decreasing or vice versa.

Try This Confirm that has a point of inflection at (0,0). No inflection point at (0,0)

Definition 2nd Derivative Test:Let f be a function such that f’(c)=0 and f” exists: If f’’(c)>0, then f(c) is a local min If f’’ (c)<0, then f(c) is a local max If f” (c)=0, test fails

Procedure Second Derivative Test: 1. Find critical numbers by setting f’(x)=0. 2.Find f’’(c) where c is a critical number. 3. f ”(c)>0 local min; f ”(c)<0 local max.

Example Find the relative extrema (max and/or min) of: using the second derivative test

Try This Find the relative extrema (max and/or min) of: using the second derivative test

Solution Max at (1,2) What did you do to determine there was no extrema at (0,0) since f ”(0)=0? Min at (-1,-2)

Lesson Close How do you test for concavity? To test for local extrema, do you prefer the 1st derivative or 2nd derivative test? Why?

Assignment 195/1-5 odd, 21-35 odd