Question

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

• A. $25$
• B. $43$
• C. $62$
• D. $49$

Answer 1

Answer: (D)

We have, $P(x)=41+24x-18x^2$
$\Rightarrow \frac{dP\left(x\right)}{dx}=24-36x$ and
$\frac{d^{2}P\left(x\right)}{d{x}^2}=-36$
For maximum or minimum, we must have
$\Rightarrow \frac{dP\left(x\right)}{dx}=0$
$\Rightarrow 24-36x=0$
$\Rightarrow x=\frac{2}{3}$
Also, ${\left(\frac{d^{2}P\left(x\right)}{d{x}^2}\right)}_{x=\frac{2}{3}}=-36<0$.
So, profit is maximum when $x=\frac{2}{3}$.
Maximum profit = (Value of $P\left(x\right)$ at $x=\frac{2}{3}$)
$=41+24\times \left(\frac{2}{3}\right)-18{\left(\frac{2}{3}\right)}^2$
$=49$