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# Find the maximum profit that a company can make if the profit function is given by $P(x)=41-24x-18x^2$

Toolbox:
• Necessary condition for maximum or minimum 'p' is $\large\frac{dp}{dx}=0$
• Condition for maximum is $\large\frac{d^2p}{dx^2}$ is negative.
Step 1
$\large\frac{dp}{dx}=-24-36x=0$
$\Rightarrow x = - \large\frac{2}{3}$
Step 2
$\large\frac{d^2p}{dx^2}=-36$ is negative
Step 3
maximum profit = $p\bigg( \large\frac{-2}{3} \bigg)$$= 49$
edited May 13, 2014