MAXIMA AND MINIMA OF FUNCTIONS Lesson 2.2. Definitions Global extrema : If f(c) f(x) for all x in...
-
Upload
nicole-roach -
Category
Documents
-
view
213 -
download
0
Transcript of MAXIMA AND MINIMA OF FUNCTIONS Lesson 2.2. Definitions Global extrema : If f(c) f(x) for all x in...
MAXIMA AND MINIMA OF FUNCTIONS
Lesson 2.2
Definitions
Global extrema: If f(c) ≥ f(x) for all x in the domain of f, f(c) is the global maximum value of f. If f(c)≤ f(x) for all x in the domain of f, f(c) is the global minimum value of f.
Local extrema: If f(c) ≥ f(x) for all x in some open interval containing c, f(c) is a local maximum value of f. If f(c)≤ f(x) for all x in some open interval containing c, f(c) is a local minimum value of f.
Example 1:
Approximate the global and local maximum and minimum on each given domain for the function k defined by
k(x) = -2x4 + 3x3 + 4x2 – 5x + 5
a. Set of all real numbers:b. -1 ≤ x ≤ 1c. x < -2
Y = button
k(x) = -2x4 + 3x3 + 4x2 – 5x + 5
Set of all real numbers:-no global min, local min. at x≈ .477, k(x) ≈3.747 - local/global max. at x ≈-.865, k(x) ≈9.257, - local max. at x ≈ 1.513, k(x) ≈ 6.502
k(x) = -2x4 + 3x3 + 4x2 – 5x + 5
-1 ≤ x ≤ 1-local/global min at x≈ .477, k(x) ≈3.747 -local min at endpoint x=-1, k(x) = 9-local/global max. at x ≈-.865, k(x) ≈9.257, - local max at endpoint, x = 1, k(x) = 5
k(x) = -2x4 + 3x3 + 4x2 – 5x + 5
x < -2- there is no minimum since the function decreases without bound on the interval (-∞, -2).-There is no maximum because k(x) increases as x increases and there is no greatest value of x on this interval.
Find the extrema of f(t)=2t4 +4t + 1
Over [0,∞) - local/global min: t=0, f(t) = 1 - No local/global max.
Over (-3,1):-local/global min: t≈-.787,
f(t) ≈ -1.381-No local/global max.
Homework
Page 91 3, 5, 6, 710, 12, 14