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This is from MIT OCW single variable calculus, in a section about learning to sketch curves. The material explains that if f'(x) is negative then f(x) is decreasing - this makes sense to me geometrically, if the slope of the tangent is negative then the graph is obviously decreasing.

Now for the question:

Sketch the graph of y = x/(x+4); find the intervals on which it is increasing and decreasing and decide how many solutions there are to y = 0."

And the solution:

y = x/(x+4), y' = 4/(x+4)^2

Increasing on: -4 < x < infinity

Decreasing on: -infinity < x < -4

I can see that the solution is correct, but as far as I can tell, y' > 0 for all values of x (except x = -4).

Is there something I'm missing, algebraically? Or am I just supposed to look at the graph and infer that this is some special case of y'?

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    $\begingroup$ The function is increasing at every point. Why do you say it is decreasing in $(- \infty , -4)$? This is wrong. $\endgroup$ – Crostul Nov 13 '16 at 23:07
  • $\begingroup$ The function decreases when x < -4 E.g. -5 / (-5 + 4) = -5 / -1 = 5; don't negative numbers divided by negative numbers become positive? $\endgroup$ – Xianny Nov 13 '16 at 23:08
  • $\begingroup$ I don't get your point. You know that $f(-5)=5$, so what? $\endgroup$ – Crostul Nov 13 '16 at 23:09
  • $\begingroup$ @Xianny That´s not true. $\endgroup$ – callculus Nov 13 '16 at 23:09
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    $\begingroup$ The function is strictly increasing on any interval where it is defined. This does not mean that it is increasing on $(-5,-3)$ for example, since that interval includes point $-4$ where the function is not defined. $\endgroup$ – dxiv Nov 13 '16 at 23:12
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enter image description here

The plot shows that the function does increase for every value of $x$.

As your derivative test did.

There is no decreasing behaviour.

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    $\begingroup$ The function is strictly increasing on $(-\infty,-4)$ and $(-4,\infty)$, but it's not monotonic on $\mathbb{R} \setminus \{-4\}$. I think that might be part of OP's confusion. $\endgroup$ – dxiv Nov 13 '16 at 23:23
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    $\begingroup$ @dxiv You're surely right about the monotony. It seemed me the OP had some issues also with a ghost decreasing part, so I plotted the function to show him at least the trend! $\endgroup$ – Von Neumann Nov 13 '16 at 23:24
  • $\begingroup$ thanks - getting a computer plotted graph helps me confirm the answer sheet is either wrong or I'm misunderstanding it! that's definitely where the confusion is coming from. $\endgroup$ – Xianny Nov 13 '16 at 23:25
  • $\begingroup$ @Xianny The function does increase for every value of $x$. At $x = -4$ the function is not defined, since the limit $4^+$ and $4^-$ are different. So you can tell the function does increase but not monotonically as dxiv suggested. If you have other doubts, please write them! $\endgroup$ – Von Neumann Nov 13 '16 at 23:26

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