The term Concavity is used to describe the type of curvature the graph displays at any given point.

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1 4 4 Concavity and the Second Derivative The term Concavity is used to describe the type of curvature the graph displays at any given point. The curve of the graph is called Up at point if the graph is BOVE the line tangent to the the graph at point. The graph approaches point from above, touches Point and then curves up away from point. Up The curve of the graph is called Down at point if the graph is below the line tangent to the the graph at point. The graph approaches point from below, touches Point, and then curves away from point. Down The first derivative is used to determine if the curve is increasing or decreasing at a given point on the function. The second derivative is used to determine if the curve is concave up or concave at a given point on the function. The 4 diagrams shows the 4 different combinations of increasing or decreasing and concave up or concave that can c=occur on a curve at point. Increasing Up Increasing Down! Decreasing Up Decreasing Down Math ! Page 1 of 10! 2018 Eitel

2 E D F C G H I J K L B The graph is decreasing and concave up at point The graph is neither increasing or decreasing at point B and is concave up at point B The graph is increasing and concave up at point C The graph is increasing and concave at point D The graph is neither increasing or decreasing at point B and is concave at point E The graph is decreasing and concave at point F The graph is decreasing and concave up at point G The graph is neither increasing or decreasing at point B and is up at point H The graph is increasing and concave up at point I The graph is increasing and concave at point J The graph is neither increasing or decreasing at point B and is concave at point K The graph is decreasing and concave at point L The graph is decreasing and concave up at point G The graph is neither increasing or decreasing at point B and is up at point H The graph is increasing and concave up at point I Math ! Page 2 of 10! 2018 Eitel

3 s If a continuous function f(x) in the neighborhood of point and concavity changes at point then point is an inllecton point. Point is an Concavity changes from! Concavity changes from concave up to concave! concave to concave up at point! at point Up Up Points E, E and H are s Concavity changes from concave up to concave at point B Concavity changes from concave to concave up at point E Concavity changes from concave up to concave at point H C D B E F G H I Math ! Page 3 of 10! 2018 Eitel

4 4 Basic Models of Concavity at a Point Point is an f(x) = x 3! f(x) = x 3 concave concave up concave up concave f(x) = 3 x f(x) = 3 x increasing concave decreasing concave increasing concave up decreasing concave up!!! For a continuous function there is an inflection point beteen a relative maximum and relative minimun. Up Up Math ! Page 4 of 10! 2018 Eitel

5 2 Combinations of Increasing/Decreasing and Up or Down can produce a graph with a sharp point called a Cusp. f(x) = x 2/3!! f(x) = x 2/3 Decreasing Down Increasing Down Relative Maximum Point Relative Minimum Point Increasing Up Decreasing Up Math ! Page 5 of 10! 2018 Eitel

6 The Second Derivative Test for Concavity! If f(x) is a function whose second derivative f (x) is defined for all x on the open interval(a,b)! 1. If f (x) > 0 for all x in (a,b) then the graph of the function is concave upward in (a,b) then! 2. If f (x) > 0 for all x in (a,b) then the graph of the function is concave ward in (a,b)! Theorem: If f(c) exists and f(c) changes sign at x=c, then the point (cf(c)) is an inflection point of the graph of f. If f(c) exists at the inflection point, then f(c)=0. The steps to find the intervals where a function is concave upward or concave ward 1. Find the second derivative of the function. 2. Find the critical numbers of the second derivative.. Find the values where the second derivative is equal to 0 f (x) = 0 B. Find the values where the second derivative is undefined f (x) is undefined 3. Plot the critical numbers on an axis labeled f (x) 4. Determine if each critical number is odd or even. 5. Determine the sign of f (x) for an x value to the right of all the critical numbers. If the sign is positive the graph of the function in that interval is concave upwards. If the sign is negative the the graph of the function in that interval interval is concave wards. Label each intervals as concave up or concave. 6. Continue to label the intervals to the left as concave upwards or concave wards If the interval to the right of an odd critical number is concave up the interval to the left is concave. If the interval to the right of an odd critical number is concave the interval to the left is concave up. If the interval to the right of an even critical number is concave up the interval to the left is also concave up If the interval to the right of an even critical number is concave the interval to the left is also concave 7. List the open intervals where the function is concave up or concave using interval notation. Math ! Page 6 of 10! 2018 Eitel

7 The Second Derivative Test for n If concavity changes at x = c then the function has an at x = C If f(x) is a function whose second derivative f (x) is defined for all x on the open interval(a,b) and c is an x value in the open interval such that a < c < b 1. If f (x) > 0 for all x > c and f (x) < 0 for all x < c then the graph of the function has an at x = c f (x) < 0 (c, f(c) ) Up f (x) > 0 2. If f (x) < 0 for all x > c and f (x) > 0 for all x < c then the graph of the function has an at x = c f (x) < 0 (c, f(c) ) f (x) > 0 Math ! Page 7 of 10! 2018 Eitel

8 Example 1! Example 2 f (x) = 1 2 x4 x 3 6x f (x) = 1 12 x4 + x 3 4x 2 2 f (x) = 2x 3 3x 2 12x f (x) = 6x 2 6x 12 ( ) f (x)) = 6 x 2 x 2 f (x) = 6 x 2 ( ) ( x +1) critical numbers where f (x) = 0 x 2 = 0 x +1= 0 x = 2 x = 1 f (x) = 1 3 x3 + 3x 2 8x f (x) = x 2 + 6x 8 f (x) = x 2 ( ) ( x 4) critical numbers where f (x)) = 0 x 2 = 0 x 4 = 0 x = 2 x = 4 critical numbers where f (x) is undefined. NONE! critical numbers where f (x) is undefined NONE f (x) is concave up (, 1) ( 2, ) f (x) is concave on ( 1,2 )! f (x) is concave up on ( 2,4 ) f (x) is concave on (,2) ( 4, ) There is an inflection point at x = 1! There is an inflection point at x = 2 There is an inflection point at x = 2! There is an inflection point at x = 4 Math ! Page 8 of 10! 2018 Eitel

9 ! Example 3 f (x) = 9 70 x10/3 9 4 x4/3 f (x) = 3 7 x7/3 3x 1/3 f (x) = x 4/3 x 2/3 f (x) = x2 1 x 2/3 critical numbers where f (x) = 0 x 2 1= 0 x = ±1 critical numbers where f (x) is undefined f (x) = x2 1 x 2/3 when x = 0 is undefined x 2/3 3 = x 2 so it is an even critical number 00 f (x) is concave up on (, 1) ( 1, ) f (x) is concave on ( 1,0 ) ( 0,1 ) There is an inflection point at x = 1 There is an inflection point at x = 1 Math ! Page 9 of 10! 2018 Eitel

10 pplications 1. Wall Street reacted to the latest report that the rate of inflation is slowing." Inflation is increasing, but at a slower rate than in the past. The rate of increasing is decreasing. 2. "Infections are increasing but as the immunization program takes hold, the rate of new infections will decrease." The number of infections is increasing, but at a slower rate than in the past. The rate of increasing is decreasing. Math ! Page 10 of 10! 2018 Eitel