Questions

MAT135H5_F25_ALL SECTIONS 4.5 Preparation Check

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Which of the following statements are true and which are false? I. If 𝑓 ′ ( 𝑐 ) = 0 then 𝑓 has either a local maximum or a local minimum at 𝑥 = 𝑐 . False II. If 𝑓 ( 𝑥 ) is increasing for all 𝑥 ∈ ( − ∞ , ∞ ) , then 𝑓 ( 𝑥 ) has no inflection points. [ Select ] False True III. There exists a function 𝑓 with a local minimum at 𝑥 = 3 and 𝑓 ′ ( 3 ) ≠ 0 . [ Select ] True False IV. If a function 𝑓 ( 𝑥 ) is increasing on ( − ∞ , 5 ) and decreasing on ( 5 , ∞ ) then 𝑓 ( 𝑥 ) has a critical point at 𝑥 = 5 . [ Select ] True False

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Question restatement: Evaluate each of the four statements I–IV as true or false, given the context. I. If f′(c) = 0 then f has either a local maximum or a local minimum at x = c. II. If f(x) is increasing for all x in (−∞, ∞), then f(x) has no inflection points. III. There exists a function f with a local minimum at x = 3 and f′(3) ≠ 0. IV. If a function f is increasing on (−∞, 5) and decreasing on (5, ∞), then f has a critical point at x = 5. Option I analysis: - The statement asserts a universal rule: a zero derivative at c guarantees a local extremum at c. In fact, this is false because a stationary point (f′(c) = 0) can correspond to a local ......Login to view full explanation

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