Ch. 5 – Applications of Derivatives 5.3 – Connecting f’ and f’’ with the Graph of f.
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Transcript of 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)