Ch. 5 – Applications of Derivatives

5.3 – Connecting f’ and f’’ with the Graph of f

• Concavity: The curvature of the graph at a point– A graph is concave up at x = c if it curves upwards

(slope is increasing) at x = c• Concave up means f’(c) is increasing and f’’(c) > 0 !

– A graph is concave down at x = c if it curves downwards (slope is decreasing) at x = c• Concave down means f’(c) is decreasing and f’’(c) < 0 !

• f(x) is concave up at x = 1 but concave down at x = -1

3 2( ) 6 3f x x x x 2'( ) 3 2 6f x x x ''( ) 6 2f x x

• Point of Inflection: a point where the graph of a function has a tangent line and changes concavity– f must be continuous at a point of inflection

• So...– To find local extrema (critical points), find when...

• f(x) changes slope (from + to – , or visa versa), or• f’(x) changes sign (when f‘(x) = 0), or• f‘(x) does not exist

– To find changes in concavity (pts. of inflection), find when...• f‘(x) changes slope (from + to – , or visa versa), or• f’’(x) changes sign (when f‘’(x) = 0 or f‘’(x) does not

exist)• If f is differentiable at all points, then...

– At a maximum, the graph is always concave down– At a minimum, the graph is always concave up

• Ex: Find the local extrema of . – Find the critical points...

– f‘ = 0 when x = -2/3 and x = 1, and f’ exists everywhere, so those are the 2 critical points

– Don’t forget to plug -2/3 and 1 into f to find the y-values!– There is a local maximum at (-2/3, 130/27) and a local

minimum at (1, 5/2).– To verify the local extrema, one can use a 2nd derivative sign

chart.

– Since f’’(-2/3) < 0 and f’(-2/3) = 0, f has a local maximum at x = -2/3

– Since f’’(1) > 0 and f’(1) = 0, f has a local minimum at x = 1

2' 3 2f x x

' 10 .f pos

3 21( ) 2 4

2f x x x x

(3 2)( 1)x x

+ +2

3 1 ' 0 .f neg

'(10) .f pos-

'' 6 1f x +

1

6- '' 0 .f neg ''(1273) .f pos

• Ex: Find the local extrema and concavity of

– f‘ = 0 when x = -2 and 2– Also, f’ does not exist for , so

consider those as endpoints

– Local min @ (-2, -4) and – Local max @ (2, 4) and

2

2

1' 8 ( 2 )

2 8f x x x

x

' 10f DNE

2( ) 8f x x x

-+

' 2.5 3.4f

'(0) 8f -

22

28 0

8

xx

x

22

28

8

xx

x

2 28 x x 28 2x

8 8x or x

8 2 82

' 10f DNE

' 2.5 3.4f

8, 0

8, 0

• Ex (cont’d): Find the local extrema and concavity of– Find the 2nd derivative!

– I’ll save you some time: x = 0 is the only point of inflection because we get 2x(x2 – 12) = 0.

– f is concave up from and concave down from .

2( ) 8f x x x

+ 0 - '' 1 .f pos ''(1) .f neg

2 2

2

22

12 8 ( 2 )

1 2 8'' ( 2 )82 8

x x x xxf x

xx

3

2

2

22

2 88

88

xx x

x xxx

2 3

2 3/22

2 (8 )0

(8 )8

x x x x

xx

( 8,0) (0, 8)