Question

the derivative of a function f is given for all x by
f'(x)= x^2(x+1)^3(x-4)^2
the set for which f is a relative maximum is.
a)0,-1,4 b)-1 c)0,4 d)1 e) none of these

Answer 1

you can use the first derivative test to determine if the solutions to f ' (x) are rel. max or min. (alternatively you can use second derivative test)

Answer 2

how would that determine if it's a max/min?

Answer 3

Yeah, just take the derivative of your function, find your critical points, then test numbers in close proximity to the critical numbers to test where it increases and decreases. If it changes sign from positive to negative, it's a relative max.

Answer 4

ok how do i find the ciritical points?

Answer 5

if you solve f ' (x) = 0, the solutions are your critical points

Answer 6

so x=0, x=-1, & x=4 are the critical points?

Answer 7

Note: The provided function is f'(x) not f(x).

Answer 8

yes those are the critical points

Answer 9

now you can make a sign chart

Answer 10

okay well it doesn't change from a positive to a negative, so that means there isn't a max right?

Answer 11

note that x^2 and (x-4)^2 are both positive
but the factor (x+1)^3 does change sign