1. Maximum ??? Minimum??? How can we tell?
If f is increasing just to the left of a critical number c
and decreasing just to the right of c,
then f has a local maximum at c
If f is decreasing just to the left of a critical number c
and increasing just to the right of c,
then f has a local minimum at c

2. First Derivative Test
Let c be a critical number of a continuous function f.
If f ' (x) changes from positive to negative at c, then f has a local maximum at c.

3. First Derivative Test
Let c be a critical number of a continuous function f.
2. If f ' (x) changes from negative to positive at c, then f has a local minimum at c.

4. First Derivative Test
Let c be a critical number of a continuous function f.
3. If f ' (x) does not change sign at c, then f has no maximum or minimum at c.

5. EXAMPLE 1: f(x) = 3x2 – 4x + 13 f ′(x) = 6x – 4 6x – 4 < 0
𝑥< 2 3
𝑥> 2 3
𝑥= 2 3
(critical number)
f ′(x) < 0
f ′(x) > 0
tangent slope is negative
tangent slope is positive
𝒇 𝟐 𝟑 = 𝟑𝟓 𝟑 =𝟏𝟏.𝟕
local minimum at

6. EXAMPLE 2: f(x) = x3 – 12x – 5 f ′(x) = 3x2 – 12 3x2 – 12 = 0
Test for x < –2
3( – 2)( )
3(x2 – 4) = 0
Test for –2 < x < 2
3( – 2)( )
3(x – 2)(x + 2) = 0
x = 2 or x = –2
Critical values
Test for x > 2
3( – 2)( )
f (–2) 11
max
min
f (2) = -21 + +
– 2 2
Local maximum at value is 11
Local minimum value is -21

7. EXAMPLE 3 f(x) = x4 – x3 f ′(x) = 4x3 – 3x2 f (0) = 0 f ( ¾ ) = 0.11
Test for x < 0
( )2 (4( ) – 3)
4x3 – 3x2 = 0
x2 (4x – 3) = 0
Test for 0 < x < ¾
( )2 (4( ) – 3)
critical values
x = 0 or x = ¾
Test for x > ¾
( )2 (4( ) – 3) +
Local maximum DNE
Local minimum value is 0.11