Question

Find the maximum profit that a company can make if the profit function is given by $p\left(x\right)=41-72x-18x^2$.

Answer 1

Compute the maximum profit of the company:

Given: profit function

and

$p\text{'}\left(x\right)=-72-36x$ [ First derivative of the given function ]

$p\text{'}\text{'}\left(x\right)=-36$ [ as we can see, this is negative for all$x$; ]

So, $p\text{'}\left(x\right)=-72-36x=0$

$36x=-72$

$x=-2$ [therefore, $p\left(x\right)$ will have its maxima at $x=-2$]

The maximum profit which the company is going to make is:

$$\begin{gathered} \Rightarrow p\left(-2\right)=41-72\left(-2\right)-18{\left(-2\right)}^2 \\ \Rightarrow p\left(-2\right)=41-\left(-144\right)-72 \\ \Rightarrow p\left(-2\right)=113 \end{gathered}$$

Hence, the maximum profit is$113$.