Section 5.1 – Increasing and Decreasing Functions The First Derivative Test (Max/Min) and its documentation 5.2

The Theory First…… THE FIRST DERIVATIVE TEST If c is a critical number and f ‘ changes signs at x = c, then f has a local minimum at x = c if f ‘ changes from neg to pos. f has a local maximum at x = c if f ‘ changes from pos to neg

_ + _ There is a rel min at x = 1 because f ‘ changes from neg to pos There is a rel max at x = 3 because f ‘ changes from pos to neg NO CALCULATOR

The Theory…Part II EXTREME VALUE THEOREM If a function f is continuous on a closed interval [a, b] then f has a global (absolute) maximum and a global (absolute) minimum value on [a, b]. GLOBAL (ABSOLUTE) EXTREMA A function f has: A global maximum value f(c) at x = c if f(x) < f(c) for every x in the domain of f. A global minimum value f(c) at x = c if f(x) > f(c) for every x in the domain of f.

The Realities….. On [1, 8], the graph of any continuous function HAS to Have an abs max Have an abs min

+ _ There is an abs min at x = -1/2

+ _ _ + Justify your answer. -2 3

+ _

Justify your answer. _ + _

+ _

GRAPHING CALCULATOR REQUIRED

x = x = x = 0

[0, 0.398), (1.351, 3]

The absolute max is and occurs when x = 3 The absolute min is –0.098 and occurs when x = 1.351

Let k = 2 and proceed

3 6 _ + +

_ + _ 0 1

CALCULATOR REQUIRED t = 3.472