Medicine It is well known ^17 that when a person coughs,...

Medicine It is well known ${ }^{17}$ that when a person coughs, the diameter of the trachea and bronchi shrinks. It has been shown that a mathematical model of the flow of air $F$ through the windpipe is given by $F(r)=k(R-r) r^{4}$, where $r$ is the radius of the windpipe under the pressure of the air being released from the lungs, $R$ is the radius with no pressure, and $k$ is a positive constant. Find the radius at which the flow is maximum.

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Key Concepts

Calculus Optimization

Calculus optimization involves finding the maximum or minimum values of a function according to its derivative characteristics. This method is widely used to determine the best or worst-case scenario in various practical applications by locating the critical points where the derivative of the function is zero or undefined.

First Derivative Test

The first derivative test is a technique used to identify the local maxima and minima of a function by setting its first derivative to zero. Once the critical points are identified, the sign changes of the derivative around these points help determine whether each point is a maximum, minimum, or neither.

Product Rule in Differentiation

The product rule is a fundamental principle in differentiation used when differentiating a function that is the product of two or more functions. According to this rule, the derivative of a product is equal to the first function multiplied by the derivative of the second plus the second function multiplied by the derivative of the first, facilitating the calculation of derivative values essential for optimization.