What are the global and local extrema of #f(x) = x^3-9x+3 # ?

1 Answer
Nov 20, 2015

Answer:

There are no global extrema. #3+3sqrt3# is a local maximum (Which occurs at #-sqrt3#) and #3-6sqrt3# is a local minimum. (It occurs at #sqrt3#.)

Explanation:

The domain of #f# is #(-oo,oo)#.

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

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

#f'(x) = 3x^2-9# is never undefined and is #0# at #x= +-sqrt3#.

We look at the sign of #f'# on each interval.

#{: (bb "Interval", bb"Sign of "f',bb" Incr/Decr"), ((-oo,-sqrt3)," " +" ", " "" Incr"), ((-sqrt3,sqrt3), " " -, " " " Decr"), ((sqrt3 ,oo), " " +, " "" Incr") :}#

#f# has a local maximum at #-sqrt3#, which is #f(-sqrt3) = 3+3sqrt3#

and a local minimum at #sqrt3#, hich is #f(sqrt3) = 3-6sqrt3#