S skyblue New member Joined Nov 5, 2006 Messages 13 Dec 7, 2006 #1 Profit= (x^3) - 48x^2 + (576x) - 1500 x= the number of units sold. How many units will be sold to maximize the company's profit?
Profit= (x^3) - 48x^2 + (576x) - 1500 x= the number of units sold. How many units will be sold to maximize the company's profit?
S soroban Elite Member Joined Jan 28, 2005 Messages 5,584 Dec 7, 2006 #2 Re: optimization Hellom skyblue! You've never done a max/min problem before? Profit: \(\displaystyle \,P(x)\;= \:x^3\,-\,48x^2\,+\,576x\,-\,1500\) . . \(\displaystyle x\) = the number of units sold How many units will be sold to maximize the company's profit? Click to expand... Set the derivative equal to 0 and solve . . . \(\displaystyle P'(x)\:=\:3x^2\,-\,96x\,+\,576\:=\:0\) Divide by 3: \(\displaystyle \:x^2\,-\,32x\,+\,192\:=\:0\) Factor: \(\displaystyle \,(x\,-\,8)(x\,-\,24)\:=\:0\) . . and we have two roots: \(\displaystyle \:x\:=\:8,\:24\) Test these with the second derivative: \(\displaystyle \,f''(x)\:=\:6x\,-\,96\) . . \(\displaystyle f''(8)\,=\,6(8)\,-\,96\:=\:-48\;\) negative, concave down, \(\displaystyle \cap\), maximum . . \(\displaystyle f''(24)\,=\,6(24)\,-\,96\:=\:+48\;\) positive, concave up, \(\displaystyle \cup\). minimum. Therefore, for maximum profit: \(\displaystyle x\,=\,8\)
Re: optimization Hellom skyblue! You've never done a max/min problem before? Profit: \(\displaystyle \,P(x)\;= \:x^3\,-\,48x^2\,+\,576x\,-\,1500\) . . \(\displaystyle x\) = the number of units sold How many units will be sold to maximize the company's profit? Click to expand... Set the derivative equal to 0 and solve . . . \(\displaystyle P'(x)\:=\:3x^2\,-\,96x\,+\,576\:=\:0\) Divide by 3: \(\displaystyle \:x^2\,-\,32x\,+\,192\:=\:0\) Factor: \(\displaystyle \,(x\,-\,8)(x\,-\,24)\:=\:0\) . . and we have two roots: \(\displaystyle \:x\:=\:8,\:24\) Test these with the second derivative: \(\displaystyle \,f''(x)\:=\:6x\,-\,96\) . . \(\displaystyle f''(8)\,=\,6(8)\,-\,96\:=\:-48\;\) negative, concave down, \(\displaystyle \cap\), maximum . . \(\displaystyle f''(24)\,=\,6(24)\,-\,96\:=\:+48\;\) positive, concave up, \(\displaystyle \cup\). minimum. Therefore, for maximum profit: \(\displaystyle x\,=\,8\)
