1. 4.3 – Derivatives and the shapes of curves
The Mean Value Theorem:
If f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that: b a c

2. Increasing / Decreasing Test
If f’(x) > 0 on an interval, then f is increasing on that interval.
If f’(x) < 0 on an interval, then f is decreasing on that interval.
ex: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and decreasing…
f ’(x) = 12x3 – 12x2 – 24x
Factor: 12x(x – 2)(x + 1)
Critical Numbers: x = -1, 0, 2
Test: ( - , -1) (-1, 0) (0, 2) (2, )

3. The First Derivative Test
For c being a critical number of a continuous function f…
If f ’ changes from positive to negative at c, then f has a local maximum at c.
If f ’ changes from negative to positive at c, then f has a local minimum at c.
If f ’ does not change sign at c, then f has no local maximum or minimum at c. c2 c1
f ’
FROM – TO + = MIN
FROM + TO – = MAX

4. ex:
Critical Numbers: x = -1, 0, 2
Test: ( - , -1) (-1, 0) (0, 2) (2, ) -1 0 2
Concavity Test
If f ’’(x) > 0 in [a,b], then f is concave up on [a,b]
If f ’’(x) < 0 in [a,b], then f is concave down on [a,b]

5. The Second Derivative Test
For f ’’ being continuous near c…
If f ’(c) = 0 and f ’’(c) > 0 then f has a local minimum at c.
If f ’(c) = 0 and f ’’(c) < 0 then f has a local maximum at c.
f ’
f ’’ c1 c2 f

6. ex: For the curve f(x) = x4 - 4x3 use calculus to find local maxima and minima, concavity intervals and points of inflection…
Critical #’s = 0, 3
[ Test f ’ ]
(- , 0)
f’(-1) =
No Max / 0
f’(1) =
(0, 3)
(3, )
f’(4) = +
NegPos =
Local Min

7. + + Test second derivative for concavity intervals: [ Test f ’’ ]
2nd derivative test f ’ critical # 0
[ Test f ’’ ]
(- , 0)
f ’’(-1) = +
f ’ critical # 3 = (36)(1) = 36, so f ’’ is + = 3 CC
Inflection 0
f ’’(1) =
(0, 2) CC
Inflection 2
(2, )
f ’’(5) = + CC
Summary
Local min at (3, -27)
No local max
CC up from (– , 0) U (2, )
CC down from (0, 2)

8. f(x) = x4 - 4x3

9. nDeriv
For f(x) = x2/3(6 – x)1/3 graph the function and its first and second derivatives….
Type f(x) into Y1
Go to Y2 line
Go to MATH [8] (nDeriv) / Press ENTER
Set nDeriv arguments to (Y1, X, X)
*Y1 in VARS ~ Y-VARS ~ FUNCTION ~ #2
Repeat on Y3 line using (Y2, X, X) for nDeriv
Graph