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]} #