The maximum profit that a company can make, if the profit function is given by $\mathrm{p}\left(\mathrm{x}\right)=41-72\mathrm{x}-18{\mathrm{x}}^2$ is:

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{p}\left(\mathrm{x}\right)=41-72\mathrm{x}-18{\mathrm{x}}^2\\ \therefore {\mathrm{p}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=-72-36\mathrm{x}\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}=-\frac{72}{36}=-2\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{p}}^{\prime \prime }\left(\mathrm{x}\right)=-36\\ \mathrm{Also}, {\mathrm{p}}^{\prime \prime }(-2)=-36<0 \end{array}$
By second derivative test, x=-2 is the point of local maxima of p.
$\begin{array}{l}\text{∴ Maximum profit }=\mathrm{p}(-2)\\ =41-72(-2)-18(-2)2\\ =41+144-72=113\end{array}$
Hence, the maximum profit that the company can make is 113 units.