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Section 3.4 – Concavity and the Second Derivative Test

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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) ?

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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

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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

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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.

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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)

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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.

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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

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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.

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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.

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

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

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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.

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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

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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

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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.

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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:

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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.

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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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