Maxima & Minima Word Problem Problem: Given the following profit-versus-production function for a certain commodity: $P=200000-x-(\frac{1.1}{1+x})^8$. Where P is the profit and x is the unit of production. Determine the maximum profit.
Solution: Taking its first derivative, $\frac{dP}{dx} = -1-8(\frac{1.1}{1+x}^7) * (\frac{-1.1}{(1+x)^2})$, then equate to $0$, the value of x would be equal to $0.371$. Then substituting it to the original equation would result to $199,999.46$ which is the maximum profit.
Question:


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*How to solve if instead, the problem asked for the minimum profit?

*In some problems, the minimum is the value of x (example: the 0.371 in the problem above) after differentiating the given equation and equating it to 0. But in some problem the minimum is the value after substituting that x, so in some problem, that 199,999.46 is the minimum instead. So how can I know which is which?


Any help or tip would be appreciated.

By Second Derivative test we can decide function is maximum or minimum at point x. http://mathworld.wolfram.com/SecondDerivativeTest.html
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-maxmin-2009-1.pdf
