Where a lake gets deeper more and more slowly

For the depth to increase at a decreasing rate, the graph has to be going up and also bending downward (concave down), so each step to the right gains less height than the step before.

Read the graph of $W$ in pieces: - From $t=0$ to $t=3$, $W$ rises from 2 to about 3 and the curve gets steeper. That is increasing at an increasing rate (concave up). - At about $t=3$ the bending changes. From $t=3$ to $t=6$ the curve still rises, to its peak of 4 at $t=6$, but it flattens out as it approaches the peak. That is increasing at a decreasing rate. - From $t=6$ to $t=18$ the depth falls, so nothing there counts. - From $t=18$ to $t=30$ the depth rises again, and it gets steeper the whole way to 7. That is increasing at an increasing rate.

So the only interval is $(3,6)$, choice A.

Each choice

• A (correct). Between $t=3$ and the peak at $t=6$ the depth is going up but leveling off, which is rising and concave down.
• B. The interval runs past the peak at $t=6$. From 6 to 12 the depth is falling, so the function is not increasing there at all. Spotting where the curve is concave down without checking its direction leads here.
• C. These are the stretches where the depth rises and gets steeper. That is increasing at an increasing rate, the opposite of what was asked. Mixing up concave up and concave down produces this choice.
• D. Here the student listed every interval where the depth is increasing and ignored what the rate is doing. Only part of $(0,6)$ is concave down, and none of $(18,30)$ is.