Question

Check work please !
How can I find where a function is decreasing in interval notation if I have no critical points
f(x)=(7x-2)/(x+7)
f'(x)=47/(x+7)^2
I found that its increasing at (-inf,-7)U(-7,inf) but I don't understand how to tell where it is decreasing

Answer 1

f'(x) = 47/(x+7)^2

can this be negative ever ?

Answer 2

assuming your math for first derivative is correct, what is the second? Do you use first or second deriv. to find dec. or inc.?

Answer 3

I used the first derivative

Answer 4

it could never be negative ?

Answer 5

that means, the slope can never be negative

Answer 6

wat does that tell about the function ?
the function is always increasing right ?

Answer 7

may not be "strictly increasing", but it does tell u that the function is never decreasing

Answer 8

The function is alwas increasing, I put none as my answer but apparently its wrong since I thought the function only increased

Answer 9

You can check for critical pts on the first deriv using the second, maybe go back and check the original problem. Did you copy it correctly? Is the first deriv correct?

Answer 10

so let's redo the first deriv.

Answer 11

Can you tell me the quotient rule?

Answer 12

Okay I used quotient rule
f(x)=(7x-2)/(x+7)
f'(x)=(x+7)(7)-(7x-2)(1)/ (x+7)^2
and I would end up with actually 51/(x+7)^2

Answer 13

bottom * derivative top - top *derivative bottom, all over bottom squared ?

Answer 14

yep

Answer 15

Okay (x+7)*(7) - (7x-2)*(1)

Answer 16

So, you were right we get \(g'(x)=\frac{51}{(x+7)^2}\)

Answer 17

now, we know that assuming x is \(\underline{\color{red}{real}}\), this can never be negative.

Answer 18

BUT: We do have a singularity

Answer 19

Namely x=-7 As you pointed out above with your note of where it is increasing. @dnova21, is this for like a lon cappa or online system?

Answer 20

online system

Answer 21

so we decided there is no where where this is decreasing right?

Answer 22

When you type it in, try doing an empty set

Answer 23

Oh okay let me check

Answer 24

It worked thank you so much

Answer 25

np, I know how crappy those systems can be