1. Maxima and minima of functions
Lesson 2.2

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.

3. Example 1:
Y = button
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
Set of all real numbers:
-1 ≤ x ≤ 1
x < -2

4. 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) ≈

5. 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

6. 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.

7. 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) ≈
-No local/global max.

8. Homework
Page 91 3, 5, 6, 7 10, 12, 14