Question

Find the amount s of advertising that maximizes the profit \(P\). (s and \(P\) are measured in thousands of dollars.) Find the point of diminishing returns. \(P=-2 s^{3}+35 s^{2}-100 s+200\)

Short answer

First, differentiate the profit function to find its critical points where the maximum profit will be achieved. The solutions are found by setting the derivative equal to zero and solving the resulting quadratic equation. Check for maxima, minima, and points of inflection by plugging the solutions into the second derivative of the profit function. The point of diminishing returns is found at the value of s where the second derivative of the profit function is less than or equal to zero.

Step-by-step solution

Differentiation of Profit Function

Differentiate the profit function \( P=-2 s^{3}+35 s^{2}-100 s+200 \) with respect to \( s \) (amount of advertising). This will give you \( P' = -6s^{2} + 70s - 100 \).

Find Critical Points

Set the derivative \( P' \) to zero to find the critical points. This results in the equation \( -6s^{2} + 70s - 100 = 0 \). Solving this quadratic equation will give you the values of \( s \) at which \( P \) reaches a maximum or minimum.

Applying the Quadratic Formula

The solutions of the quadratic equation are given by the quadratic formula \( s = [-b ± sqrt(b^{2} - 4ac)] / 2a \), where \( a = -6 \), \( b = 70 \), and \( c = -100 \). Applying the formula gives \( s = [ -70 ± sqrt((70)^{2} - 4*(-6)*(-100)) ] / (2*(-6)) \). Simplifying this will give two solutions for \( s \).

Checking for Maxima, Minima, and Points of Inflection

Take the second derivative of the profit function \( P \), which gives \( P'' = -12s + 70 \). If the value of \( P'' \) is greater than zero at a critical point, then that critical point is a minima; if it's equal to zero, it's a point of inflection; and if it's less than zero, then it's a maxima. By substituting the solutions of \( s \) into \( P'' \), you can ascertain whether the points are maxima, minima, or points of inflection.

Identify the Point of Diminishing Returns

The point of diminishing returns is the point where the second derivative of the profit function is less than or equal to zero. This will be the point after which additional advertising will not increase the profit as much. Therefore, the solution to \( P'' \leq 0 \) will give the point of diminishing returns.

Key concepts

Critical Points

Critical points play a crucial role in understanding the behavior of a function. In the context of profit maximization, identifying these points helps determine where the profit function reaches its highest or lowest values.

A critical point occurs where the first derivative of the function is zero or does not exist. For our problem, we differentiate the profit function with respect to the amount of advertising, which is represented by variable \(s\).

Set the first derivative equal to zero to locate the critical points. This helps identify at what levels (or amounts) of advertising the profit may be maximized or minimized. However, the critical points alone do not tell us if they are maxima or minima. Additional analysis is necessary to make this determination, which is done using the second derivative.

Differentiation

Differentiation is a mathematical process that helps us understand how a function changes at any given point. In our profit maximization exercise, we begin by differentiating the profit function to find where the maximum profit occurs.

The profit function is given as \(P = -2s^3 + 35s^2 - 100s + 200\). Differentiating this with respect to \(s\), the amount spent on advertising, gives the first derivative \(P' = -6s^2 + 70s - 100\).

This derivative indicates the rate of change of profit with respect to advertising. By setting \(P'\) equal to zero, we can find the critical points which suggest potential maximum or minimum profit levels. In essence, differentiation simplifies understanding the relationships and patterns in complex functions by finding slopes and changes.

Diminishing Returns

The concept of diminishing returns is crucial when analyzing business strategies like advertising. It refers to a point at which the rate of profit gain is reduced with additional investment or input, signaling a less efficient outcome.

For the profit function at hand, diminishing returns can be determined by the point where the second derivative becomes non-positive. In practical terms, this is where the additional money spent on advertising no longer provides a proportionate increase in profit.

In our exercise, after finding the critical points using the first derivative, the second derivative \(P'' = -12s + 70\) helps us identify the point at which returns diminish. By solving \(P'' \leq 0\), we pinpoint when increasing advertising expenditure results in lower gains, an important factor in budget allocation.

Quadratic Formula

The quadratic formula is a tool used to solve second-degree polynomial equations, which appear frequently in optimization problems. In our exercise, once we derive the first derivative, we are left with a quadratic equation that needs solving to find the critical points.

The equation \(-6s^2 + 70s - 100 = 0\) emerges from setting the derived function equal to zero. Using the quadratic formula \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -6\), \(b = 70\), and \(c = -100\), we find the values of \(s\) that lead to possible maxima or minima of profit.

These calculations yield two solutions, which allow for further analysis using the second derivative to determine their nature (whether they maximize, minimize, or otherwise do not affect profit). Understanding these solutions is critical in identifying the optimal levels of advertising.