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

$\begin{array}{1 1} 59\\ 49 \\ 60 \\100 \end{array}$

Toolbox:
• $\large\frac{d}{dx}$$(x^n)=nx^{n-1} Step 1: p(x)=-18x^2-24x+41 p'(x)=-36x-24 \qquad=-12(2+3x) For maxima and minima p'(x)=0 Now p'(x)=0 \Rightarrow 12(2+3x)=0 2+3x=0 3x=-2 x=\large\frac{-2}{3} Step 2: p'(x) changes sign from +ve to -ve. \Rightarrow p(x) has a maximum value at x=\large\frac{-2}{3} Maximum profit=p(\large\frac{-2}{3}\big) \qquad\qquad\quad\;\;=41-24\big(\large\frac{-2}{3}\big)$$-18\big(\large\frac{-2}{3}\big)^2$
$\qquad\qquad\quad\;\;=41+16-8$
$\qquad\qquad\quad\;\;=49$