After much research, Palmoil, a mixture of palm oil (P) and diesel (D), was created. A liter of palm oil costs \$0.30 and a liter of diesel costs \$0.50. However, for a liter of Palmoil with $x \in [0,1]$ liters of palm oil the performance is $\frac{10}{1+x}$ kilometers per liter.
Determine the amount of palm oil (P) that minimizes the cost per kilometer .
My doubt is that when I create the equation I can not find minimum but just maximum.
Here is what I am doing so far: $\frac{\textrm{km}}{\textrm{l}}=(\frac{10}{1+x}).\frac{1}{x}$
cost per kilometer =$((\frac{10}{1+x}).\frac{1}{x})$.
Price=$((\frac{10}{1+x}).\frac{1}{x})(0,3x+(1-x)0,5)=\frac{-2x+5}{x+x^2}$
So the $x=\frac{5}{2}$ is a max price not minimum price. My conclusion is so far that this mixture is completely wrong. Am I right?
Your notation is a bit confusing... so I could have some inputs wrong, please correct me if there are mistakes.
We wish to optimize $x$, the palm oil fraction in Palmoil. We are given that:
(1) For each liter of Palmoil, we get $\frac{10}{1+x}$ km of performance.
(2) Since palm oil costs \$0.3/liter and diesel costs \$0.5/liter, the cost of Palmoil is $0.3x + 0.5(1-x)$ dollars per liter.
Thus, the cost per kilometer of Palmoil with palm oil fraction $x$ is given by:
$$ c(x) = \left[ \frac{1+x}{10} \frac{\textrm{L}}{\textrm{km}} \right] \left[ \left(\frac{3x}{10} + \frac{1-x}{2}\right) \frac{$}{L} \right] \\ = \left[ \left(\frac{1+x}{10}\right) \frac{\textrm{L}}{\textrm{km}} \right] \left[ \left(\frac{5-2x}{10} \right) \frac{$}{L} \right] \\ = \left( \frac{(1+x)(5-2x)}{100}\right) \frac{$}{\textrm{km}} \\ = \frac{1}{100} (5 + 3x - 2x^2) \frac{$}{\textrm{km}}$$
We want to minimize $c(x)$. However, $c''(x) = -4/100 < 0$, so there is no local minimum. Any critical points will be a local maximum. Thus, the minimum must occur at one of the endpoints of the domain of $x$, which must be between $0$ and $1$.
$c(0) = \frac{5}{100}$ and $c(1) = \frac{6}{100}$.
Thus, the optimal cost to performance ratio is $x = 0$, which means Palmoil should just be entirely diesel... ha.