Presentation on theme: "Section 3.4 – Concavity and the Second Derivative Test"— Presentation transcript:
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) = 3x2 +3 then f(x) is always increasing because f '(x) is always positive. But which graph below represents f(x)?
3 Concavity CONCAVE DOWN CONCAVE UP 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 DOWNCONCAVE UP
4 Test for Concavity Slopes are decreasing. CONCAVE UP If f ''(x) > 0 for all x in I, then the graph of f is concave upward on I.If f ''(x) < 0 for all x in I, then the graph of f is concave downward on I.Slopes are decreasing.CONCAVE UPSlopes are increasing.CONCAVE DOWN
5 Find where the first derivative is increasing and decreasing. Procedure for Finding Intervals on which a Function is Concave Up or Concave DownIf 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:Find the critical numbers of f ' and values of x that make f '' undefined in (a,b).These numbers divide the x-axis into intervals. Test the sign (+ or –) of the second derivative inside each of these intervals.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 f is concave down when the derivative is decreasing. Use the graph of f '(x) below to determine when f is concave up and concave down.f is concave down when the derivative is decreasing.f ' (x)xf is concave up when the derivative is increasing.A critical point for the first derivative (f''=0)Concave Up:(1, ∞)Concave Down:(-∞,1)
7 Example 2Find where the graph is concave up and where it is concave down.Domain of f:All RealsFind the critical numbers of f 'Find the first derivative.Find where the 2nd derivative is 0 or undefinedFind the second derivative.Find the sign of the second derivative on each interval.Answer the questionThe function is concave down on (-∞,0) because f ''<0 and is concave up on (0,∞) because f ''>0
8 The Change in Concavity 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.CONCAVE DOWNCONCAVE UP
9 Inflection PointsA 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 Example Determine the points of inflection of . Domain of f: Find the critical numbers of f 'All RealsFind the derivative.Find where the 2nd derivative is 0 or undefinedFind the second derivative.The 2nd derivative is undefined at t=6.NOTE: 6 is not an inflection point since the 2nd derivative does not change sign.Find the sign of the second derivative on each interval.Find the value of the function:67.2Answer the question
11 Example: AnswerThe point of Inflection is (7.2,8.131) because the second derivative changes from negative to positive values at this point.
12 White Board ChallengeFind where the inflection point(s) are for f if f ''(x) = 4cos2x – 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?The function is concave downward at B.The function is concave upward at C.Df(x)BACx
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)Relative Maximumf '(c) = 0f ''(c) < 0cx
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 '(c) = 0f ''(c) > 0Relative Minimumcx
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)Here is a common example of neither a minimum or maximumf '(c) = 0When the Second-Derivative Test fails, the critical point can often be classified using the First-Derivative Test.f ''(c) = 0cx
17 Example 1 Domain of f: Find the relative extrema of . All Reals Find the critical numbersApply the 2nd Derivative TestFind the first derivative.Find where the derivative is 0 or undefinedFAILMINMAXApply the 1st Derivative Test where the 2nd Derivative Test Fails.Find the value of the function:-11Answer the questionNo sign change means x=0 is not a relative extrema.
18 Example 1: AnswerThe 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 2The 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)?