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Consider the expression

$$y=\frac{1375\times44.5}{x}+\frac{1375}{7−0.1(x−110)−0.05n}\times1.49$$

where $n$ are integers between $0$ and $10$ to keep it simple.

How do I find the relationship between $x$ and $n$ such that $y$ will always be a minimum? Also, how would I express it with some formula?

lioness99a
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Son Jerm
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  • If you make x a very small negative number then you can make y a very large negative number. Do you really mean the absolute value of y should be a minimum? – user121049 May 05 '17 at 08:35
  • Yes, y such that it is positive. I have experimented with different n values and found that the x- coordinate of the minimum decreases as n increases. I understand this an inversely proportional relationship between x and n, but how would I interpret it and is there a way to derive an expression for the minimum that is resulted as n varies? – Son Jerm May 05 '17 at 08:45
  • if you differentiate w.r.t. $x$ and look for a minimum of y by setting $dy/dx=0$ you get a quadratic in x. Solving this will give x as a function of n for which y is either a minimum or a maximum. You will need to check that x is real, is a minimum, and y>0. – user121049 May 05 '17 at 12:37
  • Hi I've tried what you have suggested but I am getting stuck with solving $dy/dx=0$. I got the following numerator for dy/dx:$-407x^2+220275x-6118.75ax-15296.875a^2+1101375a-19824750$ but i am unable to solve it for 0. – Son Jerm May 07 '17 at 01:39
  • Plug it into the standard formula. See http://www.mathsisfun.com/algebra/quadratic-equation.html. Looks like it will be messy! – user121049 May 08 '17 at 07:17

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