Chapter 3: Applications of Differentiation

1. Chapter 3: Applications of Differentiation L3.3 Increasing / Decreasing Functions and the First Derivative Test

2. Increasing and Decreasing Functions If for every x1 < x2on an interval • f(x1) < f(x2),then f is increasing [as x→, f↑ ] • f(x1) > f(x2),then f is decreasing [as x →, f↓ ] • f(x1) = f(x2),then f is constant [as x →, f is flat] Tests for increasing and decreasing functions: • If f ’(x) > 0 for all x in (a, b), then f is increasing on [a, b]. • If f ’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]. • If f ’(x) = 0 for all x in (a, b), then f is constant on [a, b]. A function is strictly monotonic on an interval if it is either increasing or decreasing on the entire interval. f ’(x) < 0 f ’(x) > 0 decreasing increasing f ’(x) = 0 constant

3. Increasing and Decreasing Functions Functions change between ↑ and ↓ at relative extrema.These are peaks and valleys of the function’s graph. • Example: Find the open intervals on which is increasing and decreasing. f is everywhere continuous. Critical numbers? f’(x) = 3x2 – 3x. f’(x) is not undefined anywhere. f’(x) = 0 at x = 0, 1 Test intervals: Answer: f ↑ing on (−∞, 0) and (1, ∞) f ↓ing on (0, 1) – + + 1 0

4. First Derivative Test Let c be a critical number of f that is continuous on an open interval containing c. • If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f. • If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f. Any x-values not in the domain of f must also be considered when testing intervals. f ’(x) > 0 f ’(x) > 0 f ’(x) < 0 f ’(x) > 0 f ’(x) < 0 f ’(x) > 0 relative minimum relative maximum neither max nor min

5. First Derivative Test Find relative extrema for • f(x) = 3x5 – 20x3 • f(x) = sinx∙cosx on the open interval (0, 2π) Steps: Find critical numbers and domain restrictions Test f’(x) in each interval Mins occur where f’(x) changes from negative to positive Max’s occur where f’(x) changes from positive to negative