Calculus - finding maximum profit?

R(x) = (125 / x^2 - 12x + 61) + 4
E(x) = [sqrt(2x + 1)] + 3

x is hundreds of components produced,
R(x) and E(x) are in thousands of dollars

If profit is calculated as the difference between revenue and expense, P(x) = R(x) - E(x), how many items should be produced to maximize profit?

Question

Calculus - finding maximum profit?

R(x) = (125 / x^2 - 12x + 61) + 4
E(x) = [sqrt(2x + 1)] + 3

x is hundreds of components produced,
R(x) and E(x) are in thousands of dollars

If profit is calculated as the difference between revenue and expense, P(x) = R(x) - E(x), how many items should be produced to maximize profit?

anonymous · Answer

I know that I should subtract R(x) - E(x), then find the critical number from that and test to see if it's the maximum, but I seem to be helpless at the calculation (with the square root being there). Can anyone show me how to do this? Do I need to make the denominators of R(x) and E(x) the same before finding the derivative? Thanks in advance.

dumbcow · Answer

$$P(x) = \frac{125}{x^{2}-12x+61} - \sqrt{2x+1} +1$$
$$P'(x) = \frac{-125(2x-12)}{(x^{2}-12x+61)^{2}} - \frac{1}{\sqrt{2x+1}}=0$$
to solve get common denominator....then denominator goes away because it equals 0
$$(x^{2}-12x+61)^{2} = -125(2x-12)\sqrt{2x+1}$$

dumbcow · Answer

From here you could square both sides but its too complex of a polynomial really to do by hand If you have a graphing calculator, graph it to see possible zeros http://www.wolframalpha.com/input/?i=%28x%5E2+-12x+%2B61%29%5E2+%3D+-125%282x%2B12%29sqrt%282x%2B1%29

anonymous · Answer

Okay... thanks for your effort! :)