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 obtain a maximum profit. (b) The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns.

Short answer

The company should spend $40,000 on advertising to maximize profit and the point of diminishing returns is when the company spends $20,000 on advertising.

Step-by-step solution

Differentiate the Profit Function

We begin by finding the derivative of the profit function \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\), which gives us \(P'=-\frac{3}{10} s^{2}+12 s\).

Find the Maximum Point

To find the maximum value of the profit function, we set the derivative equal to zero and solve for \(s\). So, we solve \(-\frac{3}{10} s^{2}+12 s=0\). Dividing through by \(s\) (where \(s \neq 0\)) gives us \(-\frac{3}{10} s +12=0\). Solving this equation results in \(s=40\). This indicates that to maximize profit, the company should spend $40,000 on advertising.

Second Derivative Test

Now, we take the second derivative of the profit function, giving us \(P''=-\frac{6}{10} s +12\). Here, we only need the sign of the second derivative at the critical point \(s=40\). Substituting \(40\) into \(P''\), we get \(P''(40) = -\frac{6}{10} * 40 + 12 = -12 < 0\). As the second derivative at this point is less than zero, our point \(s=40\) is a maximum.

Find the Point of Diminishing Returns

The point of diminishing returns is where the rate of growth of the profit function begins to decline, thus when the second derivative \(P''\) is zero. So, we solve \(-\frac{6}{10} s +12 = 0\) which gives us \(s=20\). Therefore, the point of diminishing returns is at \(s=20\), meaning when the company spends $20,000 on advertising, the growth rate of profit begins to decline.