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

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