MATH 12002 - CALCULUS I 3.3: Concavity & Inflection Points - - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §3.3: Concavity & Inflection Points - Example

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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Example

Example

Let f (x) = 3x4 − 4x3. Determine intervals where f is increasing, intervals where f is decreasing, the location of all local maxima and minima, intervals where f is concave up, intervals where f is concave down, and the location of all inflection points.

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Example

We need to determine the signs of f ′ and f ′′ for f (x) = 3x4 − 4x3. We have f ′(x) = 12x3 − 12x2 = 12x2(x − 1), f ′′(x) = 36x2 − 24x = 36x(x − 2

3).

Hence f ′ and f ′′ are never undefined and f ′(x) = 0 when x = 0 or x = 1; f ′′(x) = 0 when x = 0 or x = 2

3.

In order to determine the signs of both f ′ and f ′′ on each interval, we put all of these numbers in our sign chart:

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Example

f (x) = 3x4 − 4x3, f ′(x) = 12x2(x − 1), f ′′(x) = 36x(x − 2

3)

2 3

1 12x2 x − 1 f ′(x) 36x x − 2

3

f ′′(x)

Inc-Dec Concave Shape

+ + + + − − − + − − − + − + + + − − + + + − + + ✲ ✲ D D D I ✛ ✛ U D U U ✍ ☞ ✍ ✌

INF INF MIN D.L. White (Kent State University) 4 / 11

Example

f (x) = 3x4 − 4x3, f ′(x) = 12x2(x − 1), f ′′(x) = 36x(x − 2

3)

2 3

1

Inc-Dec Concave Shape

D D D I

MIN

U D U U

INF INF

✡ ✟ ✡ ✠

f is increasing on (1, ∞), f is decreasing on (−∞, 1); f has a local minimum at x = 1. f is concave up on (−∞, 0) ∪ (2

3, ∞), f is concave down on (0, 2 3);

f has an inflection point at x = 0 and at x = 2

3.

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Example

In order to sketch the graph of f , we will need to plot the points whose x coordinates are in the sign chart. We need to evaluate f (x) = 3x4 − 4x3 at these points: f (0) = 3(04) − 4(03) = 0 f (2

3)

= 3( 2

3)4 − 4(2 3)3

= ( 2

3)3

3(2

3) − 4

• =

( 8

27)(2 − 4)

= ( 8

27)(−2) = −16 27 ≈ −0.59

f (1) = 3(14) − 4(13) = 3 − 4 = −1 Hence the points (0, 0), (2

3, −16 27), and (1, −1) are on the graph.

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Example

2 3

1

Inc-Dec Concave Shape

D D D I

MIN

U D U U

INF INF

✡ ✟ ✡ ✠ ✲ ✛ ✻ ❄

1 −1 −1 1 2

q

(0, 0), INF

q

( 2

3 , − 16 27 ), INF

q

(1, −1), MIN D.L. White (Kent State University) 7 / 11

Example

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Example

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Example

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Example

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