What are the global and local extrema of #f(x)=x^3-3x+6# ?

1 Answer
Jim H
Dec 3, 2015

That function has no global extrema. It has local maximum of #8# (at (#x=-1#) and local minimum of #4# (at #x=1#)

Explanation:

#lim_(xrarroo)f(x) = oo#, so there is no global maximum.

#lim_(xrarr-oo)f(x) = -oo#, so there is no global minimum.

#f'(x) = 3x^2-3# which is never undefined and is #0# at #x=-1# and at #x=1#. The domain of #f# is #RR#.

Therefore, the only critical numbers are #-1# and #1#.

The sign of #f'# changes from + to - as we pass #x=-1#, so #f(-1) = 8# is a local maximum.

The sign of #f'# changes from - to + as we pass #x=1#, so #f(1) = 4# is a local minimum.

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