If the revenue function is R = Q 3Q^2 and cost function C = Q^2 2Q , find the value of maximum profit.

To find the value of maximum profit, we first need to determine the profit function $P$ by subtracting the cost function $C$ from the revenue function $R$.

Given:

• Revenue function: $R=Q-3Q^2$
• Cost function: $C=Q^2-2Q$

Step 1: Determine the Profit Function

The profit function $P$ is given by: $$P=R-C$$ Substituting the given functions: $$P=(Q-3Q^2)-(Q^2-2Q)$$ Simplify the expression: $$P=Q-3Q^2-Q^2+2Q$$ $$P=3Q-4Q^2$$

Step 2: Find the Critical Points

To find the maximum profit, we need to find the critical points by taking the derivative of the profit function with respect to $Q$ and setting it to zero: $$\frac{dP}{dQ}=\frac{d}{dQ}(3Q-4Q^2)$$ $$\frac{dP}{dQ}=3-8Q$$ Set the derivative equal to zero to find critical points: $$3-8Q=0$$ $$8Q=3$$ $$Q=\frac{3}{8}$$

Step 3: Verify the Maximum Profit

To ensure that this critical point corresponds to a maximum, we use the second derivative test. Compute the second derivative of $P$: $$\frac{d^{2}P}{d{Q}^2}=\frac{d}{dQ}(3-8Q)=-8$$ Since the second derivative is negative ($-8<0$), the function $P$ has a maximum at $Q=\frac{3}{8}$.

Step 4: Calculate the Maximum Profit

Substitute $Q=\frac{3}{8}$ back into the profit function to find the maximum profit: $$P=3\left(\frac{3}{8}\right)-4{\left(\frac{3}{8}\right)}^2$$ $$P=\frac{9}{8}-4\left(\frac{9}{64}\right)$$ $$P=\frac{9}{8}-\frac{36}{64}$$ $$P=\frac{9}{8}-\frac{9}{16}$$ $$P=\frac{18}{16}-\frac{9}{16}=\frac{9}{16}$$

Thus, the value of maximum profit is $\frac{9}{16}$.