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 83–85, you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Per-form the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where ƒ′ = 0. (In some exercises, you may have to use the numerical equation solver to ap-proximate a solution.) You may want to plot ƒ′ as well.
c. Find the interior points where ƒ′ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function’s absolute extreme values on the interval and identify where they occur.
83. ƒ(x) = x4 - 8x2 + 4x + 2, [-20/25, 64/25]
84. ƒ(x) = -x4 + 4x3 - 4x + 1, [-3/4, 3] 85. ƒ(x) = x^(2/3)(3 - x), [-2, 2]
In Exercises 15–22, calculate the approximation for the given function and interval.
For Problems 4 – 8, let S be an uncountable set. Label each of the following statements as true or false, and justify
your answer.
There exists a bijective function f : J → S.
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