How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)= 3x^4+4x^3-12x^2+5#?

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Answer 1 · 2016-12-10T04:43:41

Tthe function has a maximum at #x=0#
The function has a minimum at #x=1#
The function has a minimum at #x=2#

Given -

#f(x)= 3x^4+4x^3-12x^2+5#

find the first two derivatives

#f^'=12x^3+12x^2-24x#
#f^('')=36x^2+24x-24#

Set first derivative equal to zero

#f^' = 0=>12x^3+12x^2-24x=0#

Find the values of #x#

#12x(x^2+x-2)#
#12x(x^2+2x-x-2)#
#12x[x(x+2)-1(x+2)]#
#12x(x-1)(x+2)#

#12x=0#
#x=0#

#x-1=0#
#x=1#

#x+2=0#
#x=2#

#x # has three values

At #x=0#

#f^('')=36(0)^2+24(0)-24=-24 < 0#

At #x=0; f^'=0; f^('')<0#

Hence the function has a maximum at #x=0#

At #x=1 #

# f^('')=36(1)^2+24(1)-24#
# f^('')=36+24-24=36 > 0#

At #x=0; f^'=0; f^('')<>0#

Hence the function has a minimum at #x=1#

At #x=2#

# f^('')=36(2)^2+24(2)-24#
# f^('')=144+48-24=168 > 0#

At #x=0; f^'=0; f^('')<>0#

Hence the function has a minimum at #x=2#

graph{3x^4+4x^3-12x^2+5 [-10, 10, -5, 5]} #