Sec 4.3 – Monotonic Functions and the First Derivative Test Monotonicity – defines where a function is increasing or decreasing. A function is.
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Transcript Sec 4.3 – Monotonic Functions and the First Derivative Test Monotonicity – defines where a function is increasing or decreasing. A function is.
Sec 4.3 â Monotonic Functions and the First Derivative Test
Monotonicity â defines where a function is increasing or
decreasing.
A function is monotonic if it is increasing or decreasing on an interval.
đ đĽ
a c
d
b
Monotonicity of đ(đ)
Interval
Increasing/Decreasing
(đ, đ)
đđđđđđđ đđđ
(đ, đ)
đđđđđđđ đđđ
(đ, đ)
đđđđđđđ đđđ
Sec 4.3 â Monotonic Functions and the First Derivative Test
The First Derivative Test
A function đ đĽ is continuous on an open interval containing critical point(s). If
đ đĽ is differentiable on the interval, except possibly at the critical points, then
đ đĽ at the critical point(s) can be classified as follows:
1. Local Maximum if đď˘ đĽ changes from positive to negative at m.
2. Local Minimum if đď˘ đĽ changes from negative to positive at n.
3. If there is no sign change, then the critical point is not a local minimum or
maximum.
ďˇ
ďˇ
đ đĽ
ďˇ
a e m
f
n
g
b
Sec 4.3 â Monotonic Functions and the First Derivative Test
The First Derivative Test
ďˇ
ďˇ
đ đĽ
ďˇ
a e m
f
n
f(critical point)
Extrema
đ đ
đđđđđ đđđĽ
đ đ
đđđđđ đđđ
Test Point
fâ(test point)
fâ(x)
Inc/Dec
e
đď˘ đ = +
đď˘ đĽ > 0
Inc.
f
g
đď˘ đ = â
đď˘ đ = +
đď˘ đĽ < 0
đď˘ đĽ > 0
g
b
Dec.
Inc.
Sec 4.3 â Monotonic Functions and the First Derivative Test
Example Problems
đ đĽ = đĽ 3 â 6đĽ â 3
đ đĽ = đĽ 4 â 6đĽ 2 + 2
đ đĽ = đĽ(đđđĽ)2
đ đĽ = đĽ 3 â 4đĽ 2 + 3đ đđđĽ
Sec 4.4 â Concavity and Curve Sketching
Concavity â defines the curvature of a function.
A function is concave up on an open interval if đď˘ đĽ is increasing on the
interval.
A function is concave down on an open interval if đď˘ đĽ is decreasing on the
interval.
Point of Inflection (poi) â the point on the graph where the concavity changes.
đ đĽ
ďˇ
a
c
poi
Concavity of f(x)
b
Interval
Concave up/Concave down
(đ, đ)
đđđđđđŁđ đđđ¤đ
(đ, đ)
đđđđđđŁđ đ˘đ
Sec 4.4 â Concavity and Curve Sketching
The Second Derivative Test for Concavity
The graph of a twice-differentiable function y = f (x) is:
1. Concave up on any interval where đď˛ đĽ > 0, and
2. Concave down on any interval where đď˛ đĽ < 0.
ďˇ
ďˇ
đ đĽ
ďˇ
a e
f
g
b
x
fââ(x)
fââ(x)
Concave up/Concave down
e
đď˘ď˘ đ = â
đď˘ď˘ đĽ < 0
Concave down
f
đď˘ď˘ đ = â
đď˘ď˘ đĽ < 0
Concave down
g
đď˘ď˘ đ = +
đď˘ď˘ đĽ > 0
Concave up
Sec 4.4 â Concavity and Curve Sketching
The Second Derivative Test for Local Extrema
If đď˘ đ = 0 (which makes x = c a critical point) and đď˘ď˘ đ < 0, then f has a local
maximum at x = c.
If đď˘ đ = 0 (which makes x = c a critical point) and đď˘ď˘ đ > 0, then f has a local
minimum at x = c.
NOTE: If the second derivative is equal to zero (or undefined) then the Second
Derivative Test is inconclusive.
ďˇ
đ đĽ
ďˇ
a
Critical Point
m
n
b
fââ(x)
Concavity
Extrema
m
đď˘ď˘ đ = â
Concave down
Local max
n
đď˘ď˘ đ = +
Concave up
Local min
Sec 4.4 â Concavity and Curve Sketching
Example Problems
đ đĽ = đĽ(đđđĽ)2
đ đĽ = đĽ 3 â 4đĽ 2 + 3đ đđđĽ
đ đĽ = đĽ 4 â 6đĽ 2 + 2
đ đĽ = đĽ 3 â 6đĽ â 3
Sec 4.4 â Concavity and Curve Sketching
Curve Sketching