Drug Concentration The percent of concentration of a drug in...

Drug Concentration The percent of concentration of a drug in the bloodstream $x$ hours after the drug is administered is given by
$$K(x)=\frac{4 x}{3 x^{2}+27}$$
a. On what time intervals is the concentration of the drug increasing?
b. On what intervals is it decreasing?

Key Concepts

Increasing and Decreasing Functions

This concept involves analyzing the behavior of functions over specific intervals. A function is said to be increasing on an interval if its values rise as the input increases, and decreasing if they fall. Determining this behavior typically involves evaluating the sign of the derivative on the interval: a positive derivative indicates increasing behavior while a negative derivative indicates decreasing behavior.

Derivative

The derivative of a function is a fundamental tool in calculus that measures the rate at which the function's value changes with respect to changes in its independent variable. It is crucial for determining the increasing or decreasing nature of a function, as the sign of the derivative at a point indicates whether the function is rising or falling at that point.

Critical Points

Critical points are specific values in the domain of a function where the derivative is zero or undefined. These points are important because they often represent local extrema or transitions in the behavior of the function, such as shifts from increasing to decreasing. Analyzing these points helps determine the overall trends in the function's behavior.

Quotient Rule

The quotient rule is a method used in differentiation for functions that are expressed as one function divided by another. It provides a formula for finding the derivative of a ratio of two differentiable functions, which is particularly useful when dealing with rational functions. This rule is essential for determining the derivative of the given drug concentration function.

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