`f(x) = x^3 - 12x + 2` (a) FInd the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.
f'(x)=3x^2-12 and f''(x)=6x
(a) The function increases when f'(x)>0 and decreases when f'(x)<0, so
3x^2-12>0 ==> x^2>4 ==> x<-2 or x>2
3x^2-12<0 ==> x^2<4 ==< -2<x<2
So the function increases on (-infty,-2) and (2,infty) and decreases on (-2,2).
(b) f'(x)=0 ==> x=-2 or 2. Since the function increases to the left of -2 and decreases to the right, x=-2 is a local maximum. (Also, f''(-2)<0.) Thus x=2 is a local minimum.(f''(2)<0)
(c) f''(x)=0 ==> x=0.
For x<0 f''(x)<0 so the function is concave down.
For x>0 f''(x)>0 so the function is concave up.
x=0 is an inflection point.
(d) the graph: