3.3 Day 1: The First Derivative Test

o Theorem 3.15: Let f be continuous on [a,b] and differentiable on (a,b).

1. If '( ) 0f x > for every x in (a,b), then f is increasing on [a,b].

2. If '( ) 0f x < for every x in (a,b), then f is decreasing on [a,b].

o Finding Intervals in Which a Function is Increasing and Decreasing (Without a Calculator)

1. Find the critical values of f.

2. Draw a number line and label the critical values on the number line.

3. Use test values between the critical values and substitute them into the derivative to

see if the function is increasing (positive derivative = positive slope) or decreasing

(negative derivative= negative slope)

� Find the intervals in which the function is increasing and decreasing.

1) 3 2( ) 5 5f x x x x= + − −

Calculus: LUPO

3.3 Notes Day 1

Name: _____________________________________

2) 2( ) 6 9 5f x x x= − +

3) 3 2( ) 40 8f x x x x= − − +

4) 2 3( ) ( 5) (5 2 )f x x x= − −

First Derivative Test: Let c be a critical value for f, and suppose that f is continuous

at c and differentiable on an open interval (a,b) containing c, except possibly at c itself.

1. If 'f changes from positive to negative at c, then f(c) is a local maximum of f.

2. If 'f changes from negative to positive at c, then f(c) is a local minimum of f.

3. If '( ) 0f x > or '( ) 0f x < for every x in the interval (a,b) except c, then f(c) is not a local

extremum of f.

1a) 2a) 3a)

1b) 2b) 3b)

HW: 3.3 Day 1 Worksheet