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

1 Answer
Sep 25, 2015

Answer:

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#.