Increasing and Decreasing Functions and the First Derivative Test

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Increasing and Decreasing Functions

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Increasing and Decreasing Functions and the First Derivative Test 3.3 Increasing and Decreasing Functions and the First Derivative Test A function is increasing on an interval if for any two numbers x1 and x2 in the interval x1 < x2 implies f(x1) < f(x2). A function is decreasing on an interval if for any two numbers x1 and x2 in the interval x1 < x2 implies f(x1) > f(x2).

If f’(x) > 0 x in (a,b), then f is increasing on (a,b). 2. If f’(x) < 0 x in (a,b), then f is decreasing on (a,b). 3. If f’(x) = 0 x in (a,b), then f is constant on (a,b). To find the open intervals on which f is increasing or decreasing, locate the critical numbers in (a,b) and use these numbers to determine the test intervals. Then determine the sign of f’(x) at one value in each of the test intervals. Use the above guidelines then to determine where f is increasing or decreasing.

Ex. 1 C.N.’s 0, 1 Now, test each interval. 0 1 inc. dec. inc. 1st der. test maximum minimum

Ex. 2 2 1 2 - dec. inc. dec.

Ex. 3 f’(-3) < 0 f’(-1) > 0 f’(1) < 0 f’(3) > 0 -2 0 2 dec. inc. dec. inc. (-2, ) (0, ) (2, ) 0 (-4)2/3 0 1st der. test min. max. min.

Ex. 4 C.N.’s 0, -1, 1 -1 0 1 dec. inc. dec. inc. 1st der. test (-1, ) (0, ) (1, ) 2 2 min neither min