For Exercises 67–73 , assume that f is differentiable over ( − ∞ , ∞ ) . Classify each of the following statements as either true or false. If a statement is false, explain why. If f has exactly two critical values at x = a and x = b , where a < b , then there must exist at least one point of inflection at x = c such that a < c < b . In other words, at least one point of inflection must exist between any two critical points.
For Exercises 67–73 , assume that f is differentiable over ( − ∞ , ∞ ) . Classify each of the following statements as either true or false. If a statement is false, explain why. If f has exactly two critical values at x = a and x = b , where a < b , then there must exist at least one point of inflection at x = c such that a < c < b . In other words, at least one point of inflection must exist between any two critical points.
Solution Summary: The author explains that if a function f has exactly two critical values, then there must be at least one point of inflection between the two points.
For Exercises 67–73, assume that f is differentiable over
(
−
∞
,
∞
)
. Classify each of the following statements as either true or false. If a statement is false, explain why.
If
f
has exactly two critical values at
x
=
a
and
x
=
b
, where
a
<
b
, then there must exist at least one point of inflection at
x
=
c
such that
a
<
c
<
b
. In other words, at least one point of inflection must exist between any two critical points.
In Exercises 27–28, let f and g be defined by the following table:
f(x)
g(x)
-2
-1
3
4
-1
1
1
-4
-3
-6
27. Find Vf(-1) – f(0) – [g(2)]² + f(-2) ÷ g(2) ·g(-1).
28. Find |f(1) – f0)| – [g(1)] + g(1) ÷ f(-1)· g(2).
For Problems 4 – 8, let S be an uncountable set. Label each of the following statements as true or false, and justify
your answer.
Every function f : S → J is onto.
In Exercises 13-14, find the domain of each function.
13. f(x) 3 (х +2)(х — 2)
14. g(x)
(х + 2)(х — 2)
In Exercises 15–22, let
f(x) = x? – 3x + 8 and g(x) = -2x – 5.
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