Example 4: 

The profit for a company Extreme Widgets Inc. is given by:

where x is the number of widgets made and sold (in thousands) and P is the profit in thousands of dollars. However, the maximum capacity the plant has currently to produce widgets is 15 thousand widgets . Find where the profit attains an absolute minimum and an absolute maximum value. Graph the function and visually confirm you results.

1. Since f(x) is continuous, it will attain a maximum and a minimum value somewhere on the interval from x = 0 to x = 15. Since the function is a polynomial, there won't be any sharp peaks or discontinuities to be concerned about.

2. Find the derivative, set it equal to zero, and solve for x. Then find the y value at that x. What do you get?

Since we cannot have a negative number of widgets, x = 7.6 is the only critical value that makes sense. P(7.6) = 1,891 approximately, or a profit of about $1,891,000.

3. Check the endpoints. This means, find P(0) and P(15). What y values do you get at the endpoints?

f(0) = 1000. This does not quite make sense, since it says that if no widgets are produced, the profit is $1,000,000. However, upon checking further, we find that the government is attempting to help the widget industry deal with a glut of widgets, and will pay widget producers not to produce widgets. OK, so this is a little far fetched. :-) But the model tell us that we are still getting a profit, so be it.

P(15) = 250, or a profit of $250,000. We infer the costs of widget production are eating into our profits.

4. Compare your results from steps 2 and 3. What is the larges value, and what is the smallest value?

P(0) = 1000 , P(7.6) = 1891 , P(15) = 250

The absolute min is 250 at x = 15, and the absolute max is 1891 at x = 7.6. To earn a max profit, produce 7600 (7.6 thousand) widgets.

5. Check the graph. Are your results reasonable?

The graphs below verify our conclusions. 

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