Question

Find all critical points of:
f(x)= 3x^4 - 2x^3 - 12x^2 + 18x
and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.
I took the 1st derivative and got:
12x^3 - 6x^2 - 24x +18
but after this I need to find the critical points. Not sure how to do this with all of the x's

Answer 1

lets start bt taking out the common factors

12x^3 - 6x^2 - 24x +18

6(2x^3 -x^2 -4x +3)
---------------------

2x^3 -x^2 -4x +3 = 0 now we can use some trial and error techniques but improve out odds by taking the factors of the 1st and last

1, 3
---- this gives us a "pool" of options to limit our search.
1, 2

you can try regular long division, but the synthetic divisioj is more compact:


-- -----------
x=1 | 2 -1 -4 3
0 2 1 -3
-------------
2 1 -3 0 <--remainder 0, its good.

I got lucky onthat one :)

(x-1) ( 2x^2 +x -3 )

the new expression there can be fatored easily into:

(x-1)(x+3)(x-2) = 0

x = -3, 1, 2

Answer 2

ok as far as the factored equation (x-1)(x+3)(x-2)=0. It has been a while since I have done this kind of problem. Need a refresher course. The (x+3)(x-2) should equal the (2x^2 +x -3) shouldn't it?

Answer 3

you know what, your right lol ..... i forgot about my "2" there in the front to use as a divisor on this....

Answer 4

(x-1)(x+3/2)(x-2/2) = 0

(x-1)(x+3/2)(x-1) = 0

x = -1 and x = -3/2

Answer 5

....x = 1 and x = -3/2

thats my final offer :)

Answer 6

hey :P