What are the critical numbers of the function f(x) = (x+2)^3 * (x-1)^4 ?

Question

Critical numbers are obtained by setting the first derivative equal to zero.
I just want to verify my answers.

Answer 1

Yeah, Fred's right
the answer is in the relationship

(fg)' = f' g + g' f

so then you get
( (x+2)^3 * (x-1)^4 )' =

3*(x+2)^2 * (x-1)^4 + 4*(x-1)^3 * (x+2)^3

there is some common stuff in there on both sides of the + that gives you

(x+2)^2 * (x-1)^3 * ( 3*(x-1) + 4*(x+2) )

and that equals

(x+2)^2 * (x-1)^3 * (7x+5)

which is 0 at -2, 1, -5/7

I recently found this thingy that you can use to check your work

http://people.hofstra.edu/staff/steven_r_costenoble/Graf/Graf.html

just put (x+2)^3*(x-1)^4 in the box and you can see the turning points on the graph

PS: this part is true too, but doesn't show up in this case
"you need to see if the derivative is undefined for any inputs"

Answer 2

The derivative is :
3 (x+2)² (x-1)²(x-1)²+4(x-1)²(x-1)(x+2)(x+2)²
Also written :
(x-1)²(x-1) (x+2)² [3 (x-1) + 4 (x+2)]
Or
(x-1)²(x-1) (x+2)² (7x+5)
So this derivative will be equal to 0 for the values :
x= 1
x= -2
x = -5/7

Have a nice day

Answer 3

-2, +1

Answer 4

not only does the derivative at the critical numbers need to be 0, it could also be undefined, so you need to see if the derivative is undefined for any inputs.