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)

$49$

Answer 2

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 $x = \frac{2}{3}$) $= 41+24 \times \left(\frac{2}{3}\right) - 18 \left(\frac{2}{3}\right)^{2}$ $= 49$