How do you use the first and second derivatives to sketch #y = x^3 - 12x - 12#?

1 Answer
Feb 1, 2016

See the answer bellow.

Explanation:

#y=x^3-12x-12#
First derivative: #y'=3x^2-12#
Second derivative: #y''=6x#

Using #y'=3x^2-12#
#3(x^2-4)#
#3(x-2)(x+2)#
#x=-2,2#
Sub x values into #y=x^3-12x-12#, to find y values
#y=(-2)^3-12(-2)-12#
#y=4#, #(-2,4)#
#y=(2)^3-12(2)-12#
#y=-28#, #(2,-28)#

Using #y''=6x#
Sub x values into #y''=6x#
#6(-2)=-12#
#x<0#
#:.# #(2,-28)# is a maximum point
#6(2)=12#
#x>0#
#:.# #(-2,4)# is a minimum point

When #x=0#
#y=0^3-12(0)-12#
#y=-12#
#(0,-12)#

Use the information from the first and second derivative to sketch the graph as well as the y intercept.

graph{y=x^3-12x-12 [-10, 10, -5, 5]}