Question

Given f ′(x) = (5 – x)(8 – x), determine the intervals on which f(x) is increasing or decreasing.

a)Decreasing (–∞, 5); increasing on (8, ∞)
b)Increasing (–∞, –5) U (–8, ∞); increasing on (–5, –8)
c)Decreasing (5, 8); increasing on (–∞, 5) U (8, ∞)
d)Decreasing (–∞, 5) U (8, ∞); increasing on (5, 8)

Answer 1

Please try to solve the inequality below:
\[\left( {5 - x} \right)\left( {8 - x} \right) \geqslant 0\]

Answer 2

do you want me to factor it out to get:
40-5x-8x+x^2
40-13x+x^2

Answer 3

@Nnesha @ganeshie8 @iambatman @EclipsedStar

Answer 4

@Compassionate

Answer 5

@Kainui

Answer 6

for example, try to solve this:
5-x >=0

Answer 7

oh so x= -5 and x = -8

Answer 8

no, those are the roots, you have to find the solution of the inequality (5-x)(8-x)>=0, instead

Answer 9

i dont see where the > comes in place with the = sign

Answer 10

here i'll give it a shot:
5-x>=0
-x>=-5
x>=5

Answer 11

there is a theorem which states this:
" a function f(x) is an increasing function if and only if f '(x) >=0"
So you have to apply that theorem

Answer 12

do you mean greater than or equal to 0?

Answer 13

yes!

Answer 14

the last step is incorrect since if you multiply an inequality by a negative quantity, then that inequality change its direction, so if we have

-x >= -5
then
x <= 5

Answer 15

ohhh i thought you had them as 2 separate things,
perfect :) yes im sorry i forgot that rule
so i dont see how it relates to the answer

Answer 16

it is simple, please solve this another inequality:
8-x >=0

Answer 17

x<=8

Answer 18

i think i figured it out! is it C? :)

Answer 19

that's right
so, we have this subsequent drawing:

Answer 20

wait please!

Answer 21

c)Decreasing (5, 8); increasing on (–∞, 5) U (8, ∞)

Answer 22

do you unerstand that diagram?

Answer 23

yes it is increasing

Answer 24

oops...understand...

Answer 25

on (5,8)

Answer 26

that diagram tell us that when x<=5 both quantities namely (5-x) and (8-x) are positive, so their product is also positive

Answer 27

got it

Answer 28

now please look at this diagram:

Answer 29

that diagram tell usthat whe x is grether than 8 both quantities, namely (5-x) and (8-x) are neative, nevertheless their product is stil positive.

Answer 30

oops ...are negative...

Answer 31

its okay :) with that being said, i think its b because positive means increase correct?

Answer 32

please wait
our conclusion is that the function f'(x) is positive when x <5 and x >8
so applying the theorem above, our function f(x) is an icreasing function when x<5 and x>8

Answer 33

yes! first derivative positive, means increasig function

Answer 34

okay so is it b? :) b)Increasing (–∞, –5) U (–8, ∞); increasing on (–5, –8)

Answer 35

I think that ur function is an incresing function in
\[\left( { - \infty ,5} \right) \cup \left( {8, + \infty } \right)\]

Answer 36

ohh true cause it would be the opposite decreasing on both positive 5 & 8 (5,8)
:) THANKU!!

Answer 37

thank you!!

Answer 38

U r an amazing teacher!!!! :D

Answer 39

thank! again!