Question

f'(x)=(x + 1)^2(x − 4)^3(x − 5)^4 and i have to find largest open interval on which f is increasing. i found the f'(-2)=-ve . f'(2)=-ve , f'(4.5)=+ve and f'(6)=+ve

Answer 1

the function is increasing from the 4 to infinity, is that right? (4,INFINITY)

Answer 2

I would plot the "zeros" of f' (i.e. plot the x values where f ' = 0) which you can "read off"
When f ' is zero, the parent function f is at a max, min, or inflection point.

Answer 3

f'=0 is at -1,4,5

Answer 4

we should list the sign of f ' between the zeros.
x< -1 f' (x) < 0
x = -1 f' = 0
-1 < x < 4 f ' < 0
x = 4 f ' =0
4 < x < 5 f ' > 0
x = 5 f ' = 0
5< x f ' > 0
at x=5 the function has a slope of 0, which I interpret as not increasing.
for x>5 f' is positive, meaning the slope is is positive and the parent function f is increasing.

Answer 5

I think the curve starts rising at x=4
reaches an inflection point at x=5
then starts rising again

so from 4 to infinity it never *decreases*
but I think that is different from always increases
it always increases from 5

5 < x < infinity

Answer 6

yeah i wrote 5 before hand and got it wrong

Answer 7

since its increasing at 4 to 5 aswell can i just say (4,INFINITY)

Answer 8

yes, I would try (4,infty)

I had my doubts on this one... it seemed like they are being tricky. Unless they define what they mean by increasing, you could legitimately answer 2 different ways.
I don't think mind reading (what are they *really* asking) should be part of math.

Answer 9

it just states "State the largest open interval on which f is increasing."