Question

How do use the first derivative test to determine the local extrema `f(x) = (x+3)(x-4)^2`?

Answer 1

See the explanation.

Explanation:

`f(x) = (x+3)(x-4)^2`

`f'(x) = [1] (x-4)^2 + (x+3)[2(x-4)(1)]`

`= (x-4)[(x-4)+(x+3)2]`

`= (x-4)(3x+2)`

The critical numbers for `f` are `4 " and "-2/3`

We look at the sign of `f'` on each interval to determine whether `f` is increasing or decreasing on the interval

#{: (bb "Interval", bb"Sign of "f',bb" Incr/Decr"), ((-oo,-2/3)," " +" ", " "" Incr"), ((-3/2,4), " " -, " " " Decr"), ((4,oo), " " +, " "" Incr") :}#

`f` has a local maximum of `1372/27` at `-2/3`

and a local minimum of `0` at `4`.