How do you determine whether the function #f(x)=x^3+3x^2+5x+7# is concave up or concave down and its intervals?

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Answer 1 · 2015-07-26T16:54:52

Investigate the sign of the second derivative, #f''(x)#

For #f(x)=x^3+3x^2+5x+7#, we have

#f'(x)=3x^2+6x+5# and

#f''(x) = 6x+6#

#f''(x) = 0# at #x= -1#, so we check the sign of #f''# on each side of #-1#

#f''(x)# is negative (that is #f''(x) < 0#) for #x < -1#. So the graph of #f# is concave down on #(-oo, -1)#.

#f''(x) >0# for #x > -1#, so the graph of #f# is concave up on #(-1,oo)#

The point #(-1, 4)# is the inflection point for the graph.