Concavity And The Second Derivative Test Worksheet With Answers Page 2
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9nd
Calculus Maximus
WS 3.4: Concavity & 2
Deriv Test
(
)
( )
−∞ ∞ such that the equation
f x =
_____ 5. Let f be a twice-differentiable function on
,
0
has exactly
3 real solutions, all distinct. Consider the following possibilities:
( )
′
=
I. the equation
f x
0
has at most 3 distinct roots.
( )
′′
= has at least 1 root.
II. the equation
f
x
0
( )
III. the function
f x must be a polynomial function of degree 3.
Which of these properties will f have?
(A) I only
(B) II only
(C) I and II only (D) III only (E) I, II, and III
[
]
−
_____ 6. Let f be a continuous function on
5,3
with a vertical
x = − , horizontal tangents at
x = − and
tangent line at
1
3
x = and a cusp at
x = − . The graph of f is given at
1
2
right. Which of the following properties are satisfied?
( )
(
)
′′
< on
−
I.
f
x
0
2,1
II. f has exactly 2 local extrema
III. f has exactly 4 critical points
(A) I only (B) II only
(C) III only
(D) II and III only
(E) I, II, and III
_____ 7.
x
0
2
4
6
( )
′′
f
x
−
2
6
0
2
The polynomial function f has selected values of its second derivative f ′′ given in the table
above. Which of the following statements must be true?
(
)
(A) f is increasing on the interval
2,6 .
(
)
(B) f is decreasing on the interval
2,6 .
x = .
(C) f has a local maximum at
4
x = .
(D) The graph of f has a point of inflection at
4
(
)
(E) The graph of f changes concavity in the interavl
2,6 .
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