Question

The profit function is $P\left(x\right)=41+24x-18x^2$, calculate the maximum profit of the company.

Answer 1

$P\left(x\right)=41+24x-18x^2$ ...........(i)
diff w.r. to $x$
$P^{\prime }\left(x\right)=24-36x$ ........(ii)
$24-36x=0$
$-36x=-24$
$x=\frac{24}{36}=\frac{2}{3}$
$x=\frac{2}{3}$
Again diff. w.r. to $x$
$P^{\prime \prime }\left(x\right)=-36$ which is negative
Hense $P\left(x\right)$ is maxima at $x=\frac{2}{3}$
$P\left(\frac{2}{3}\right)=41+24\times \frac{2}{3}-18\times {\left(\frac{2}{3}\right)}^2$
$P\left(\frac{2}{3}\right)=41+16-18\times \frac{4}{9}$
$P\left(\frac{2}{3}\right)=41+16-8$
$P\left(\frac{2}{3}\right)=49$.