Behaviors of Implicit Relations (College Board AP® Calculus AB)

A particular type of medication is given to a patient. The amount, in milligrams, of the medication in the patient at time hours is modeled by a function that satisfies the differential equation . At time hour, there are milligrams of the medication in the patient. Is the rate of change of the amount of medication in the patient increasing or decreasing at ? Give a reason for your answer.

Consider the curve given by the equation . It can be shown that .

Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.

Evaluate at the point on the curve where and .

Consider the curve with equation .

The point lies on the curve. Investigate whether this point is a critical point; and if it is, classify it as either a relative maximum point or a relative minimum point.

Consider the function whose curve is given by the equation for .

It is known that .

For and , find the coordinates of the point where the line tangent to the curve is horizontal.

Determine whether has a relative minimum, a relative maximum, or neither at the point found in part (a). Justify your answer.