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.
Suppose that ff"' is continuous and f'(c) =f"(c) =0 , but f"'(c) >0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c?
If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I containing c, then what can we say about f(c)?
Suppose that ƒ(x) = x2 and g(x) = | x | . Then thecompositions( ƒ ∘ g)(x) = | x | 2 = x2 and (g ∘ ƒ)(x) = | x2| = x2 are both differentiable at x = 0 even though g itself is not differentiable at x = 0. Does this contradict the Chain Rule? Explain.
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