How do you use the first derivative to determine where the function #f(x)= 3 x^4 + 96 x# is increasing or decreasing?

1 Answer
Nov 25, 2016

Answer:

#f(x)# is decreasing when #x in ] -oo,-2 ] #
#f(x)# is increasing when #x in[-2, +oo[#

Explanation:

#f(x)=3x^4+96x#

The derivative is,
#f'(x)=12x^3+96#

#f'(x)=0#

when, #12x^3+96=0#

#12x^3=-96#

#x^3=-8#

#x=-2#

Let's do a sign chart,

#color(white)(aaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaaaa)##-2##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##f'(x)##color(white)(aaaaaa)##-##color(white)(aaaaa)##0##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##darr##color(white)(aaa)##-144##color(white)(aaaa)##uarr#

So, #f(x)# is decreasing when #x in ] -oo,-2 ] #

and #f(x)# is increasing when #x in[-2, +oo[#