Thank you for reporting, we will resolve it shortly
Q.
Find the maximum profit that a company can make, if the profit function is given by $P(x) = 41 + 24x - 18x^2$.
Application of Derivatives
Solution:
We have, $P(x) = 41 + 24x - 1 8x^2$
$\Rightarrow \frac{dP\left(x\right)}{dx} = 24 - 36x$ and
$\frac{d^{2}P\left(x\right)}{dx^{2}} = -36$
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, $\left(\frac{d^{2}P\left(x\right)}{dx^{2}}\right)_{x = \frac{2}{3}} = -36 < 0$.
So, profit is maximum when $x = \frac{2}{3} $.
Maximum profit = (Value of $P\left(x\right)$ at $x = \frac{2}{3}$)
$= 41+24 \times \left(\frac{2}{3}\right) - 18 \left(\frac{2}{3}\right)^{2}$
$= 49$