What are the local extrema of #f(x)= x^3 - 9x^2 + 19x - 3 #?

1 Answer
Feb 2, 2017

Answer:

#f(x)_max =(1.37, 8.71)#
#f(x)_min =(4.63, -8.71)#

Explanation:

#f(x)= x^3-9x^2+19x-3#

#f'(x) = 3x^2-18x+19#

#f''(x) = 6x-18#

For local maxima or minima: #f'(x) =0#

Thus: #3x^2-18x+19 =0#

Applying the quadratic formula:

#x=(18+-sqrt(18^2-4xx3xx19))/6#

#x=(18+-sqrt96)/6#

#x=3+-2/3sqrt6#

#x~= 1.367 or 4.633#

To test for local maximum or minimum:

#f''(1.367) < 0 -># Local Maximum

#f''(4.633) > 0 -># Local Minimum

#f(1.367) ~= 8.71# Local Maximum
#f(4.633) ~= -8.71# Local Minimum

These local extrema can be seen on the graph of #f(x)# below.

graph{ x^3-9x^2+19x-3 [-22.99, 22.65, -10.94, 11.87]}