How do I find the critical values to find the maximum of this function?

Question

The total daily profit in dollars realized by the TKK Corporation in the manufacture and sale of x dozen recordable DVDs is given by the total profit function below. $$P(x) = −0.000001x^3 + 0.001x^2 + 5x − 430$$ $$0 ≤ x ≤ 2000$$ Find the level of production that will yield a maximum daily profit. (Round your answer to the nearest integer.) So for $P'$ I have $P'(x)=-.000003x^2+.002x+5$ And after switching the signs and multiplying by 1,000,000 to turn each decimal into integers I end with $P'(x)=3x^2-2,000x-5,000,000$. And after applying the quadratic formula I got $1666.66666667$ which rounded up is correct and I got the right answer, but I was wondering if there was an easier way without dealing with the huge number I used. Thanks in advance.

Answer 1

To avoid the large numbers (though I don't know why you want to), you could look at the ratios of the coefficients. They get smaller by a factor of about $100$ for each power of $x$. That suggests that the natural variable is $\frac 1{100}x$. If you rewrite the function in terms of $x=100y$ you get $P(y)=y^3+10y^2+500y-430$ and the numbers will be smaller. When you round, you need to round to $0.01$ This approach can get fooled if the high order term(s) is(are) a small correction.