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Section 3.4 – Concavity and the Second Derivative Test.

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1 Section 3.4 – Concavity and the Second Derivative Test

2 The Second Derivative and the Function The first derivative tells us where a function is increasing or decreasing. But how can we tell the manner in which a function is increasing or decreasing? For example, if f '(x) = 3x 2 +3 then f(x) is always increasing because f '(x) is always positive. But which graph below represents f(x) ?

3 Concavity If the graph of a function f lies above all of its tangents on an interval I, then it is said to be concave up (cupped upward) on I. If the graph of a function f lies below all of its tangents on an interval I, then it is said to be concave down (cupped downward) on I. CONCAVE DOWN CONCAVE UP

4 Test for Concavity a)If f ''(x) > 0 for all x in I, then the graph of f is concave upward on I. b)If f ''(x) < 0 for all x in I, then the graph of f is concave downward on I. Slopes are decreasing. Slopes are increasing. CONCAVE DOWN CONCAVE UP

5 Procedure for Finding Intervals on which a Function is Concave Up or Concave Down If f is a continuous function on an open interval (a,b). To find the open intervals on which f is concave up or concave down: 1.Find the critical numbers of f ' and values of x that make f '' undefined in (a,b). 2.These numbers divide the x-axis into intervals. Test the sign (+ or –) of the second derivative inside each of these intervals. 3.If f '' (x) > 0 in an interval, then f is concave up in that same interval. If f ''(x) < 0 in an interval, then f is concave down in that same interval. Find where the first derivative is increasing and decreasing.

6 Example 1 Use the graph of f '(x) below to determine when f is concave up and concave down. (1, ∞) Concave Up: Concave Down: (-∞,1) f is concave down when the derivative is decreasing. f ' (x) x f is concave up when the derivative is increasing. A critical point for the first derivative (f''=0)

7 Example 2 Find where the graph is concave up and where it is concave down. Find the first derivative. Find the critical numbers of f ' Find where the 2 nd derivative is 0 or undefined 0 Find the sign of the second derivative on each interval. Answer the question The function is concave down on (-∞,0) because f '' 0 Domain of f : All Reals Find the second derivative.

8 If a graph changes from concave upward to concave downward (or vice versa), then there must be a point where the change of concavity occurs. This point is referred to as an inflection point. The Change in Concavity CONCAVE DOWN CONCAVE UP

9 Inflection Points A point P on a curve is called an inflection point if the graph is concave up on one side of P and concave down on the other side. In calculus terms, (if f is continuous on an interval that contains c ) c is an inflection point if f '' changes from positive to negative or vice versa at c. Thus, f '' (c) must equal 0 or be undefined.

10 Find the value of the function: Example Determine the points of inflection of. Find the derivative. Find the critical numbers of f ' Find where the 2 nd derivative is 0 or undefined 6 Find the sign of the second derivative on each interval. Answer the question Domain of f : All Reals Find the second derivative. The 2 nd derivative is undefined at t= NOTE: 6 is not an inflection point since the 2 nd derivative does not change sign.

11 Example: Answer The point of Inflection is (7.2,8.131) because the second derivative changes from negative to positive values at this point.

12 White Board Challenge Find where the inflection point(s) are for f if f ''(x) = 4cos 2 x – 2 on [0,π]. Justify.

13 How the concavity is connected to Relative Minimum and Maximum When a critical point is a relative maximum, what are the characteristics of the function? When a critical point is a relative minimum, what are the characteristics of the function? f(x)f(x) x D A C B The function is concave downward at B. The function is concave upward at C.

14 The Second Derivative Test Let f be a function such that f '(c) = 0 (a critical number of a continuous function f(x) ) and the second derivative exists on an interval containing c. (a) If f '' (c) < 0, there is a relative maximum at x = c. f(x)f(x) xc f '' (c) < 0f ' (c) = 0 Relative Maximum

15 The Second Derivative Test Let f be a function such that f '(c) = 0 (a critical number of a continuous function f(x) ) and the second derivative exists on an interval containing c. (b) If f '' (c) > 0, there is a relative minimum at x = c. f(x)f(x) xc f '' (c) > 0f ' (c) = 0 Relative Minimum

16 The Second Derivative Test Let f be a function such that f '(c) = 0 (a critical number of a continuous function f(x) ) and the second derivative exists on an interval containing c. (c) If f '' (c) = 0, then the second-derivative test fails (either a maximum, or a minimum, or neither may occur). f(x)f(x) x f ' (c) = 0 Here is a common example of neither a minimum or maximum c f '' (c) = 0 When the Second- Derivative Test fails, the critical point can often be classified using the First- Derivative Test.

17 Example 1 Find the relative extrema of. Find the first derivative. Find the critical numbers Find where the derivative is 0 or undefined 1 0 Apply the 1 st Derivative Test where the 2 nd Derivative Test Fails. Answer the question Domain of f : All Reals Apply the 2 nd Derivative Test FAIL MIN MAX No sign change means x=0 is not a relative extrema. Find the value of the function:

18 Example 1: Answer The function has a relative maximum of 4 at x = -1 because the second derivative is negative at this point. The function has a relative minimum 0 at x = 1 because the second derivative is positive at this point.

19 Example 2 The Second Derivative Test is less popular than the First Derivative Test for two reasons: the Second Derivative Test does not always work and a question often requires a student to find where a function is increasing/decreasing before asking for the relative min/max (Thus, the sign chart for the First Derivative Test is done already). Yet, the AP Test will have questions, like the one below, specifically about the Second Derivative Test: Example: If the function f has a horizontal tangent line at x = 4 and f ''(4) = 3, what is true about f(4) ?


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