Answer to Question #17679 in Calculus for hsd

Question

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.

Answer 1

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