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#?
1 Answer
Tthe function has a maximum at
The function has a minimum at
The function has a minimum at
Explanation:
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
#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#
At
#f^('')=36(0)^2+24(0)-24=-24 < 0#
At
Hence the function has a maximum at
At
# f^('')=36(1)^2+24(1)-24#
# f^('')=36+24-24=36 > 0#
At
Hence the function has a minimum at
At
# f^('')=36(2)^2+24(2)-24#
# f^('')=144+48-24=168 > 0#
At
Hence the function has a minimum at
graph{3x^4+4x^3-12x^2+5 [-10, 10, -5, 5]} #