How do you find the exact relative maximum and minimum of the polynomial function of #f(x) = x^3-6x^2+15#?

1 Answer
Mar 13, 2016

Relative maximum: #(0,15)#
Relative minimum: #(4,-17)#

Explanation:

Relative maximums and minimums occur whenever the derivative equals 0. Using the power rule, we find the derivative is:
#f'(x)=3x^2-12x#

Setting it equal to #0# yields:
#0=3x^2-12x#
#0=x^2-4x#
#0=x(x-4)#
#x=0# and #x=4#

To find the exact minimum and maximum, we evaluate these two #x#-values (called critical points):
#f(0)=(0)^3-6(0)^2+15=15-># maximum
#f(4)=(4)^3-6(4)^2+15=64-96+15=-17-># minimum

Looking at the graph of the function, we can see that there is indeed a local max at #(0,15)# and a local min at #(4,-17)#.
graph{x^3-6x^2+15 [-42.95, 49.5, -19.6, 26.67]}