# Finding intervals on the graph of a function

I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows:

The rate of r (in liters per minute) at which water is entering a tank is given for t > 0 by the graph. A negative rate means that water is leaving the tank. In parts (a) – (d), give the largest interval on which: (a) The volume of water is increasing. (b) The volume of water is constant. (c) The volume of water is increasing fastest. (d) The volume of water is decreasing.

From my point of view it seems that the largest interval on which the function is increasing is also the largest interval on which it is increasing fastest. It also seems the largest intervals on which the function remains constant are the same length. Am I on the right track or am I misinterpreting the question?

Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.