题目
题目

MAT135H5_F25_ALL SECTIONS 4.5 Preparation Check

多重下拉选择题

a) Suppose we have the following information about a continuous function 𝑓 ( 𝑥 ) . Interval: ( − ∞ , 2 )   ( 2 , ∞ )   sign of 𝑓 ( 𝑥 )   - - sign of 𝑓 ′ ( 𝑥 ) : + - sign of 𝑓 ″ ( 𝑥 ) : - - Then 𝑓 ( 𝑥 ) has [ Select ] a corner or cusp a local maximum an inflection point a local minimum at 𝑥 = 2 .    b) Suppose we have the following information about a continuous function 𝑔 ( 𝑥 )   . Interval: ( − ∞ , 1 )     ( 1 , ∞ )   sign of 𝑔 ( 𝑥 )   - + sign of 𝑔 ′ ( 𝑥 ) : + + sign of 𝑔 ″ ( 𝑥 ) : + - Then 𝑔 ( 𝑥 ) has [ Select ] a local minimum a corner or cusp an inflection point a local maximum at 𝑥 = 1 .    c) Suppose we have the following information about a continuous function ℎ ( 𝑥 ) . Interval: ( − ∞ , 3 )   ( 3 , ∞ )   sign of ℎ ( 𝑥 )   + + sign of ℎ ′ ( 𝑥 )   : - + sign of ℎ ″ ( 𝑥 )   : + + Then ℎ ( 𝑥 ) has a local minimum at 𝑥 = 3  . 

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思路分析
To tackle this question, I will go part by part and assess what each given sign pattern implies about the shape of the graphs. a) For f(x) with intervals (-∞, 2) and (2, ∞), we have: sign f(x) = +, -; sign f'(x) = +, -; sign f''(x) = -, -. - Option: a corner or cusp. A corner or cusp would typically involve a sharp point where the derivative does not exist, but here we are given a defined, nonzero sign pattern for f', so the derivative exists on both sides of x = 2. Therefore a corner/cusp is unlikely. - Option: a local maximum. Since f'(x) > 0 on the left of x = 2 and f'(x) < 0 on the right, the function rises then falls at x = 2, which corresponds to a local maximum. This......Login to view full explanation

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