How do I find max profit if I am only given a cost function and how much the item sells for? $$C(X)=x{^3}-60{x^2}+1400x+1000$$ Product sells for $\$563$ per unit.
I am also given $0\leq x \leq 50$. No demand or supply function has been given. Only cost. I have already tried taking the derivative, dividing by $x$ to get the average.
I attempted to take the derivative of the cost function but then noticed its a cost function not revenue, so thats out of the bat. I also attempted to take Cbar and try to get average but then saw it asked for profit then I got confused and decided to ask for help.
Profit = Income - Cost
It costs $C(x) = x^3 - 60x^2 + 1400x + 1000$ to make $x$ items, and you earn $I(x) = 563x$ for selling $x$ items. Subtracting these, we get:
Profit: $P(x) = -x^3 + 60x^2 - 837x - 1000$ To maximize profit, we need to find where the derivative is zero.
$P'(x) = -3x^2 + 120x - 837$
This function has zeroes at $9$ and $31$. Which should we use?
The second derivative test helps identify max and min.
$P''(x) = -6x + 120$
$P''(9) > 0$, so $9$ is a minimum. $P'(31) < 0$, so $31$ is a maximum.
$P(31) = \$922$