Question

The profit \(P\) (in thousands of dollars) for a company spending an amount \(s\) (in thousands of dollars) on advertising is \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\) (a) Find the amount of money the company should spend on advertising in order to yield a maximum profit. (b) Find the point of diminishing returns.

Short answer

The company should spend $40k on advertising to maximise their profit. The point of diminishing returns, where increasing spending on advertising yields diminishing increases in profit, occurs at a spend of $20k.

Step-by-step solution

Derive the Profit Function

Differentiate the profit function \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\) with respect to \(s\). The derivative, \(P'(s)\), is the rate of change of profit with respect to advertising spending. Using the power rule for derivatives, obtain \(P'(s)= -\frac{3}{10}s^{2} + 12s\).

Find Critical Points

To maximise the profit, set the derivative \(P'(s)\) equal to zero and solve for \(s\). This means solving the quadratic equation \(-\frac{3}{10}s^{2} + 12s = 0\) for \(s\). Factoring out \(s\) gives \(s( -\frac{3}{10}s + 12) = 0\). Therefore the critical points are \(s = 0\) and \(s = 40\).

Determine the Maximum Point

Taking the second derivative of \(P\) will allow you to find whether these points are maxima, minima or points of inflection. The second derivative, \(P''(s)\), is \(-\frac{6}{10}s + 12\). Substituting \(s = 0\) and \(s =40\) into \(P''(s)\) gives \(P''(0) = 12 > 0\) and \(P''(40) = -12 < 0\). Therefore, \(s = 0\) is a local minimum and \(s = 40\) is a local maximum so the company should spend $40k to maximize profits.

Find the Point of Diminishing Returns

The point of diminishing returns is the profit amount \(s\) where the second derivative becomes negative - the profit increase rate begins to decline. For the second derivative \(-\frac{6}{10}s + 12 = 0\), solving for \(s\) gives \(s = 20\). So, after spending $20k, the company will start to see diminishing returns.

Key concepts

Critical Points in Calculus

In calculus, critical points are essential for understanding the behavior of functions. These are points where the derivative of the function is either zero or undefined, indicating potential maxima, minima, or points of inflection. To find them, one typically starts by calculating the first derivative of the function. In the context of maximizing profit, such as in the given profit function \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\), we look for values of \(s\) that make the derivative \(P'(s)\) equal to zero.

Identifying critical points provides valuable business insights. For example, if \(s\) represents the spending on advertising, finding critical points helps a company determine where changes in spending may no longer result in increased profit or might even lead to a decrease. In the provided exercise, the critical points are found to be \(s = 0\) and \(s = 40\), but only context and further testing can reveal which point is the peak of profit.

Second Derivative Test

The second derivative test is an analytical tool used to determine the nature of critical points discovered during the first derivative test. After finding the critical points, you then take the second derivative of the function. By evaluating the second derivative at the critical points, one can discern if each point is a relative maximum, minimum, or a point of inflection.

The sign of the second derivative \(P''(s)\) conveys crucial information: if \(P''(s) > 0\), the point is a relative minimum; if \(P''(s) < 0\), it is a relative maximum, and if \(P''(s) = 0\), the test is inconclusive and further analysis is needed. This step is crucial in problems like profit maximization because it helps decide where to allocate resources efficiently. In our exercise, the second derivative at \(s = 40\) is negative, confirming \(s = 40\) as the point of maximum profit, while \(s = 0\) is found to be a minimum which, in a business context, typically represents no action or investment.

Point of Diminishing Returns

The concept of the point of diminishing returns is significant in both economics and calculus. It's the stage in investment or production where the rate of return begins to decrease. This point can be ascertained by observing when the second derivative of a profit function becomes negative, indicating that the concavity of the profit curve is changing from upwards to downwards.

Understanding when and why diminishing returns occur is critical for managing investments and can prevent over-spending. By recognizing this point, companies can optimize their spending. In our given exercise, spending over $20k on advertising will lead to diminishing returns, which acts as an alert for the business to avoid excessive spending on advertising beyond this amount. It is a crucial insight that helps sustain long-term profitability by preventing resources from being spent on less effective advertising.