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
Dec 10, 2016

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

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 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]} #