Question

Find the intervals of increase and decrease for the following functions.
a) h(x)=x^3(x-1)^4
b)\[y=x \sqrt{4-x}\]

Answer 1

Have you found dh/dx

Answer 2

I tried and I got this
dh/dt= x^2(4x^2-x-3).
I'm not sure if it's right...

Answer 3

i don't know if that is it or not
i will check in a moment
you did use product rule
then power rule and chain rule right?

Answer 4

Yes

Answer 5

\[h=x^3 (x-1)^4 \\ \frac{dh}{dx}=(x-1)^4 \cdot \frac{d(x^3)}{dx}+x^3 \cdot \frac{d(x-1)^4}{dx}\]

Answer 6

so you differentiated x^3 (w.r.t. x) and got ?
so you differentiated (x-1)^4 (w.r.t. x) and got ?

Answer 7

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

Answer 8

\[\frac{dh}{dx}=(x-1)^4 \cdot 3x^2+x^3 \cdot 4(x-1)^3\]
right

Answer 9

now we can factor out the x^2(x-1)^3

Answer 10

\[\frac{dh}{dx}=x^2(x-1)^3[3(x-1)+4x]\]

Answer 11

now you need to find when dh/dx=0

Answer 12

So x= 3/7 and 0?

Answer 13

x^2=0 when x=0
x-1=0 when x=?
3(x-1)+4x=0 when x=3/7
2 out 3 correct

Answer 14

oops! The last one is x=1

Answer 15

ok cool so those are our critical numbers

Answer 16

we have 4 intervals to test

Answer 17

(-inf,0)
(0,3/7)
(3/7,1)
(1,inf)

Answer 18

select a number from each
then dh/dx for each of the numbers you picked (pick only 4 one from each interval )

Answer 19

if dh/dx>0 then h is increasing
if dh/dx<0 then h is decreasing

Answer 20

So the intervals are :
increases on (-inf,3/7) and (1,inf)
decreases on (3/7,1)
???

Answer 21

So a number from (-inf,0) is -1
\[\frac{dh}{dx}=x^2(x-1)^3(7x-3) \\ \frac{dh}{dx}|_{x=-1}=(-1)^2(-1-1)^3(7 \cdot -1 -3)=1(-2)^3(-7-3) \\ =-8(-10)=80>0 \text{ so } h \text{ is increasing on } (-\inf,0)\]
now you tested also (0,3/7)?
choose an ugly number like 2/7

Answer 22

\[\frac{dh}{dx}|_{x=\frac{2}{7}}=(\frac{2}{7})^2(\frac{2}{7}-1)^3(7 \cdot \frac{2}{7}-3) \\ =pos \cdot negative \cdot negative=pos \]

Answer 23

so i would say it is increasing on (-inf,0) U (0,3/7)

Answer 24

oh okay!

Answer 25

yep it looks good except I would separate that one interval you have

Answer 26

increases on (-inf,0) or (0,3/7) or (1,inf)
decreases on (3/7,1)