Question #17679

Let f(x)=2x^3−24x+2
Input the interval(s) on which f is increasing.
Input the interval(s) on which f is decreasing.
Find the point(s) at which f achieves a local maximum.
Find the point(s) at which f achieves a local minimum.
Find the intervals on which f is concave up.
Find the intervals on which f is concave down.
Find all inflection points.

Expert's answer

we must find f' and f''

f'=6x^2-24 f'=0 x=-2, x=2

f"=12x^2 f"=0 x=0

increase where f'>0 so at x from(-infinity, -2) and(2, +infinity) decrease where f'<0 so x from (-2 , 2) local maximum x=-2 local minimum x=2 concave up where

f"<0 concave down where f">0 12x^2>0 if x not zerox=0 is not inflection point because function does notchange type of concavity at x=0

f'=6x^2-24 f'=0 x=-2, x=2

f"=12x^2 f"=0 x=0

increase where f'>0 so at x from(-infinity, -2) and(2, +infinity) decrease where f'<0 so x from (-2 , 2) local maximum x=-2 local minimum x=2 concave up where

f"<0 concave down where f">0 12x^2>0 if x not zerox=0 is not inflection point because function does notchange type of concavity at x=0

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