Question

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

Answer 1

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`