f(x)=x2+1x2 Domain: (−∞,∞)
f′(x)=(x2+1)22x(x2+1)−2x(x2)=(x2+1)22x
f′′(x)=(x2+1)42((x2+1)2−2x(2x)(x2+1))
=(x2+1)32(x2+1−4x2)=(x2+1)32(1−3x2) Find the point(s) of inflection
f′′(x)=0=>(x2+1)32(1−3x2)=0
1−3x2=0
x1=−33,x2=33 If x<−33,f′′(x)<0,f(x) is concave downward.
If −33<x<33,f′′(x)>0,f(x) is concave upward.
If x>33,f′′(x)<0,f(x) is concave downward.
f(x) is concave upward on (−33,33).
f(x) is concave downward on (−∞,−33)∪(33,∞).
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