You have noticed that paying higher wages attracts more productive employees. however, you are concerned that there may be a limit to this relationship. some experimentation has convinced you that the relationship between daily wages paid (x) and profits can be modelled by the equation profit = 50x − 0.5x2 + .001x3 + 200. the range of wages you are willing to consider is from $0 to $500 per day. determine the level of wages that will maximize profit.

The level of wages that maximize tge profit is $61.257

And the correspond value for the profit is:

Explanation:

For this case we have the following function:

Where x represent the daily wages paid , and p(x) the profit, the objective is maximize this function, and in order to do this the first step is derivate the function respect to x and we got this:

As we can see we have a quadratic equation now we need to set up equal the derivate obtained to 0 and then solve for the critical points, like this:

We can use the quadratic formula given by:

And for this case a=0.003 , b=-1 , c =50

Replacing this we got :

And we got:

Now we need to find the second derivate, like this:

And we can replace the values obtained:

So then 61.257 is a maximum.

So then 272.076 is a minimum.

So then the level of wages that maximize tge profit is $61.257

And the correspond value for the profit is:

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