Review

1
Review

• Rolles Theorem
• Suppose f(x) is continuous on a, b and
  differentiable on (a, b). If f(a)f(b), then
• there is at least one number c in (a, b) at which
  f (c)0.
• The Mean Value Theorem
• Suppose f(x) is continuous on a, b and
  differentiable on (a, b). Then
• there is at least one number c in (a, b) at which
  
• f (c)(f(b)-f(a))/(b-a)

2
Review

• Mathematical Consequence
• If f (x)0 at each point of an open interval (a,
  b), then f(x)C for all x in (a, b), where C is a
  constant.
• If f (x)g (x) at each point of an open
  interval (a, b), then f(x)g(x)C for all x in
• (a, b), where C is a constant.

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4.3 Monotonic Functions and The First Derivative
Test

• Increasing Functions and Decreasing Functions
• Definition
• Let f be a function defined on an interval I and
  let x1 and x2 be any two points in I.
• If f(x1)ltf(x2) whenever x1 lt x2 , then f is said
  to be increasing on I.
• If f(x1)gtf(x2) whenever x1 lt x2 , then f is said
  to be increasing on I

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Monotonic Functions

• A function that is increasing or decreasing on I
  is called monotonic on I.
• 2. Examples
• a) f(x)2x1 is increasing on (-8, 8)
• b) f(x)x2 is decreasing on (-8, 0) and
  increasing on (0, 8).

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The First Derivative Test

• The First Derivative Test for Monotonic Functions
• Suppose that f is continuous on a, b and
  differentiable on (a, b).
• If f(x)gt0 at each point x in (a, b), then f is
  increasing on a, b.
• If f(x)lt0 at each point x in (a, b), then f is
  increasing on a, b.

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Using The First Derivative Test

• Examples
• Find where the function is increasing or
  decreasing.
• f(x)x2-4x2
• f(x)2x-4
• 2x-4gt0, xgt2 , so f(x) is increasing on (2, 8).
• 2x-4lt0, xlt2 , so f(x) is decreasing on
• (-8, 2).

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• b) f(x)2x3-3x2-12x18
• f(x)6x2-6x-126(x2-x-2)6(x1)(x-2)
• Intervals xlt-1 -1ltxlt2 xgt2
• sign of f f(-2) f(0)
  f(3)
• 24gt0 -12lt0
  24gt0
• Behavior
• of f increasing decreasing increasing

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First Derivative Test for Local Extrema

• First Derivative for Local Extrema
• Suppose that c is a critical point of a function
  f .
• If f changes sign from negative to positive at
  c, then f has local minimum at c.
• If f changes sign from positive to negative at
  c, then f has local maximum at c.
• If f does not changes sign at c, then f has no
  local extrema at c.

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Using the First Derivative Test for Local Extrema

• Find local extrema.
• f(x)2x3-24x
• Solution f (x)6x2-246(x2-4)6(x2)(x-2)
• Critical points -2, 2.
• Intervals xlt-2 -2ltxlt2 xgt2
• sign of f f(-3) f(0)
  f(3)
• 30gt0 -24lt0
  30gt0

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Example

• So f has a local maximum at x-2, that is
• f(-2)2(-2)3-24(-2)-164832
• f has a local minimum at x2, that is
• f(2)2(2)3-24(2)16-48-32.

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Practice

• 10, 12 on page 266.

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4.4 Concavity and Curve Sketching

• Concavity
• 1) Definition
• The graph of a differentiable function f is
• Concave up on an open interval I if f is
  increasing on I
• Concave down on an open interval I if f is
  decreasing on I

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Examples

• 2). Examples
• (a) f(x)x2 is concave up on (-8, 8).
• (b) f(x)- x2 is concave down on (-8, 8).
• (c) f(x) x3 is concave down on (-8, 0) and up on
  (0, 8).

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The Second Derivative Test for Concavity

• 3). The Second Derivative Test for Concavity
• Let yf(x) be twice-differentiable on an interval
  I.
• (a) If fgt0 on I, then the graph of f is
  concave up.
• (b) If flt0 on I, then the graph of f is
  concave down.

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More Example

• Find where the function is concave up or concave
  down.
• f(x)2x3-24x
• Solution f (x)6x2-24
• f(x)12x
• 12xgt0, xgt0 , so f(x) is concave up on (0, 8).
• 12xlt0, xlt0 , so f(x) is concave down on
• (-8, 0).

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Points of Inflection

• 2. Points of Inflection
• 1). Definition
• A point where the concavity changes is called
  a point of inflection.
• 2). Examples
• f(x) x3 is concave down on (-8, 0) and up on
  (0, 8) therefore has a point of inflection at
  x0.

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• 3). To fine inflection points, solve f(x)0
  for x and then check the concavity near those
  points.
• 3. Second Derivative Test for Local Extrema
• 1) If f (c) 0 and f (c)lt0, then f has a
  local maximum at xc.
• 2) If f (c) 0 and f (c)gt0, then f has a
  local minimum at xc.
• 3) If f (c) 0 and f (c)0, then the test
  fails. The function f may have a local maximum,
  a local minimum, or neither.

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Examples

• Find the local extrema of
• f(x)x3-6x29x1
• Solution f (x)3x2-12x9, f (x)6x-12
• f (x)0, 3x2-12x90, 3(x2-4x3)0
• (x2-4x3)0, (x-1)(x-3)0, x1, x3
• f (1)6(1)-12-6lt0, f has a local maximum at
  x1, that is f(1)(1)3-6(1)29(1)15
• f (3)6(3)-126gt0, f has a local minimum at
  x3, that is f(3)(3)3-6(3)29(3)127-542711.

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Curve Sketching

• 4. Curve Sketching
• Identify the domain of f and any symmetries the
  curve may have
• Calculate f (x) and f (x)
• Find the critical points of f, and determine the
  local extrema
• Find where the function is increasing and where
  it is decreasing.
• Find the points of inflection, if any occur, and

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Curve Sketching

• determine the concavity of the curve.
• 6) Identify any asymptotes
• 7) Plot key points, such as the x- or y-
  intercepts and the points found in previous
  steps, and sketch the curve.

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Example

• Sketch the graph of f(x)x3-6x29x1
• Solution 1) The domain of f is the set of all
  real numbers.
• 2) Calculate f and f .
• f (x)3x2-12x9, f (x)6x-12
• 3) Find critical points.
• f (x)0, 3x2-12x90, 3(x2-4x3)0
• (x2-4x3)0, (x-1)(x-3)0, x1, x3

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• f (1)6(1)-12-6lt0, f has a local maximum at
  x1, that is f(1)(1)3-6(1)29(1)15
• f (3)6(3)-126gt0, f has a local minimum at
  x3, that is f(3)(3)3-6(3)29(3)127-542711.
• 4) Increasing and decreasing
• Interval xlt1 1ltxlt3 xgt3
• Sign of f -
  
• Behavior ? ?
  ?

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• 5) Inflection points.
• f (x)0, 6x-120, x2
• f(x)lt0 if xlt2 and f (x)gt0 if xgt2
• So f has an inflection point at x2.
• f(2))(2)3-6(2)29(2)18-241813.
• 6) no assymptotes (polynomial functions dont
  have any asymptotes)
• 7) Plot points and sketch the curve.

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Practice

• Sketch the graph of f(x)3x3-9x1.