Question

how to find critical points of following function,
f(x)=x^4+4x^3+6x^2+4x

Answer 1

find f'
set f'=0

Answer 2

then solve for x

Answer 3

f'(x) = 4x^3 + 12x^2 + 12x + 4

4x^3 + 12x^2 + 12x + 4 = 0

Solve for x.

Answer 4

x=-1 is one critical number
now i would use synthetic division to find the other critical numbers

Answer 5

yes, but how do we decide if its local maximum or local minimum......(it is concave function)

Answer 6

Find the second derivative.

Answer 7

or you could use the first derivative
and see if f is increasing or decreasing before and after each critical number
if goes from increasing to decreasing then you have a max
if goes from decreasing to increasing then you have a min

Answer 8

how about hessian matrix?

Answer 9

after diff. it reduces to (x+1)^3=0