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Chapter 2: Problem 78

The profit \(P\) (in hundreds of dollars) that a company makes depends on the amount \(x\) (in hundreds of dollars) the company spends on advertising according to the model \(P=230+20 x-0.5 x^{2} . \quad\) What \(\quad\) expenditure \(\quad\) for advertising will yield a maximum profit?

Short Answer

Expert verified
Hence, the maximum profit is obtained when the company spends $2000 (20 hundreds of dollars) on advertising.

Step by step solution

01

Finding the Derivative

The first action is to compute the derivative of the function. The derivative of a function can show rate of change and it is useful for understanding where the maximum value of a function is. The derivative of the function \(P=230+20x-0.5x^2\) is \(P'=20-1.0x\).
02

Setting the derivative to zero

The critical points of a function can be found by setting the derivative to zero. Therefore, we want to solve the equation \(20-1.0x=0\). Solving for \(x\) gives \(x=20\).
03

Verifying that this provides the maximum

In this case, we know there is a maximum at \(x=20\) by the fact that the coefficient in front of \(x^2\) is negative (-0.5), suggesting a parabola that opens downwards. Therefore, the maximum point (vertex) will be at the critical point we found, \(x=20\).

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