Answer
a) The effectiveness appears to be increasing from $t=0$ to $t=3$, or the moment it enters the bloodstream until 3 hours later.
The derivative at those times is positive.
b) The drug reaches its maximum effectiveness at $t=3$, or 3 hours after the drug enters the bloodstream. The derivative there is $0$.
As $t$ increases from $t=2$ to $t=3$, the derivative is still positive, but decreases.
Work Step by Step
a) The effectiveness appears to be increasing during the time when the curve goes up, which is from $t=0$ to $t=3$, or the moment it enters the bloodstream until 3 hours later.
Since the curve goes up from $t=0$ to $t=3$, the tangent lines at these points within the interval $(0,3)$ will have their slopes making acute angles with the $x$-axis. This tells us that these slopes are positive, hence the derivatives in these effectiveness-increasing times are also positive.
b) The drug reaches its maximum effectiveness at the peak of the curve, which is $t=3$, or 3 hours after the drug enters the bloodstream.
Since $t=3$ is the peak of the curve, the tangent line there is parallel with the $x$-axis, meaning that its slope is $0$, and the derivative there is $0$ as well.
As shown in a), from $t=2$ to $t=3$, the derivative is positive. But we witness here another event: the curve gets less steeper as $t$ increases, meaning the slopes, and the derivative also, decrease as $t$ increases.
