If is differentiable on then which of the following must be true?

has a maximum value on the open interval .
for some such that .
for some such that .

Question

If  is differentiable on  then which of the following must be true?

has a maximum value on the open interval .
  for some  such that  .
 for some  such that  .

BlackTom AI · Accepted Answer

Here is what we can analyze given the question as stated, noting that the answer choices themselves are incomplete in the prompt.
First, consider option I, which says the function has a maximum value on the open interval. In general, a differentiable function on an open interval need not attain a maximum there; it can approach larger values toward the endpoints without ever achieving a maximum inside the interval. For example, a function that increases toward the right endpoint without bound on an unbounded open interval would not have a maximum. Therefore, statement I does not have to be true in all cases.
Second, turn to option II, which asserts that f'(x) = 0 for some x in the interval. This aligns with Fermat's theorem: if a differentiable function has a local extremum (in particular, a maximum or minimum) at an interior point of the domain where the derivative exists, then the derivative at that point must be zero. Since the statement says there is a maximum value on the open interval, and the maximum occurs at some interior point, the derivative there must be zero. Hence, II must be true under the given condition.
Third, examine option III, which appears to claim something about the derivative being zero for some wider or different condition (the exact text is unclear in the prompt). Without a precise formulation, it's hard to affirm it as necessarily true. However, since II already captures the standard interior-extremum implication (derivative zero at the extremum), and I can fail in general, III is not guaranteed solely from differentiability on the open interval.
In summary, the only statement that must be true under the standard interpretation of the given condition is II, because differentiability together with an interior maximum enforces f'(c) = 0 at the maximizing point. The other statements fail in general: I can fail because a maximum on an open interval need not exist, and III cannot be validated without a precise phrasing and additional assumptions.

MATH_1225_17255_202501 4.2 The Mean Value Theorem

If is differentiable on then which of the following must be true? has a maximum value on the open interval . for some such that . for some such that .

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