The maximum profit that a company can make if the profit function is given by

$P(x)=32+24x-18x^2$ is

40

The correct answer is Option (2) → 40

Profit function, $P(X)=32+24x-18x^2$

$⇒P'(x)=24-36x$

for critical points,

$24-36x=0$

$x=\frac{24}{36}=\frac{2}{3}$

and,

$P''(x)=-36$ which indicates that the function is concave down and at $x=\frac{2}{3}$ [maximum].

$∴P(\frac{2}{3})=32+24×\frac{2}{3}-18×\frac{4}{9}$

$=48-8=40$