How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #2x^3 + 3x^2 - 12x#?

1 Answer
Dec 5, 2016

#f(x)# is increasing in the interval #(-oo,-2)#, reaches a maximum for #x=-2# then decreases in the interval #(-2,1)#, reaches a minimum in #x=1#, then increases indefinitely.

Explanation:

You determine intervals of increasing and decreasing and the relative maxima and minima by studying the first derivative of the function:

#f'(x) = 6x^2+6x-12#

First, we wind the points where #f'(x)=0# that are the critical points:

#6x^2+6x-12 = 0#

#x^2+x-2 = 0#

The roots are:

#x_1=-2#
#x_2 = 1#

Then we look at the sign of #f'(x)#: as this is a second order polynomial with positive leading coefficient, we know that it is negative inside the interval between the two roots, and positive outside, that is:

#f'(x) > 0# for #x in (-oo,-2)# and #x in (1,+oo)#

#f'(x) < 0# for #x in (-2,1)#

So we can state that #f(x)# is increasing in the interval #(-oo,-2)#, reaches a maximum for #x=-2# then decreases in the interval #(-2,1)#, reaches a minimum in #x=1#, then increases indefinitely.

graph{2x^3+3x^2-12x [-8, 8, -40, 40]}