5.2 first and second derivative test

1. Relative Extrema
First Derivative Test - FDT
Second Derivative Test - SDT

2. Relative Extrema
1x 2x 3x 4x
d
e
c
r
e
a
s
i
n
g
d
e
c
r
e
a
s
i
n
g
increasing
relative
maximum
relative
maximum
relative
minimum
relative
minimum

3. Critical Points
Critical points are points at which:
•Derivative equals zero (also called
stationary point).
•Derivative doesn’t exist.

4. First Derivative Test
Let f be a differentiable function with f '(c) = 0, then:
•If f '(x) changes from positive to negative, then f has a
relative maximum at c.
•If f '(x) changes from negative to positive, then f has a
relative minimum at c.
•If f '(x) has the same sign from left to right, then f
does not have a relative extremum at c.

5. Practice Time!!!
Use First Derivative Test to find critical
points and state whether they are
minimums or maximums.fx()=3x
5
3
-15x
2
3
f'x()=5x
2
3
-10x
-1
3
=5x
-
1
3
x-2( )f'x()=3´
5
3
x
2
3
-15´
2
3
x
-1
3
5x
-
1
3
x-2( )=0
5x-2( )
x
1
3
=0 Critical points
0 2(stationary)
··
0 2
+
+
__
relative
maximum
relative
minimum

6. Second Derivative Test
Suppose that c is a critical point at which f’(c) = 0,
that f(x) exists in a neighborhood of c, and that f(c)
exists. Then:
• f has a relative maximum value at c if f”(c) < 0.
•f has a relative minimum value at c if f”(c) > 0.
•If f(c) = 0, the test is not conclusive.
Note: Second derivative test is still
used to calculate max and min

7. Practice Time again !!!
Use second derivative test to find extrema
off(x)=3x
5
-5x
3
f'(x)=15x
4
-15x
2
15x
4
-15x
2
=0 15x
2
x
2
-1( )=0
critical points = 0, -1, 1
f"(x)=60x
3
-30x
f"(0)=0
f"(-1)=-30<0
inconclusive
f has a maximum at x = -1
f"(1)=30>0
f has a minimum at x = 1

8. Find extrema of

A.Using first derivative test
B.Using second derivative test
f(x)=
x
2
x
4
+16