The profit $P(x)$ is defined as the difference between revenue $R(x)$ and cost $C(x)$:

$P(x) = R(x) - C(x)$

Step 1: Substitute the functions $R(x)$ and $C(x)$:

$P(x) = (-4x^2 + 31x) - (11x + 29)$

Simplify: $P(x) = -4x^2 + 31x - 11x - 29$ $P(x) = -4x^2 + 20x - 29$

Step 2: Find the value of $x$ that maximizes $P(x)$.

The profit function $P(x)$ is a quadratic equation, and it opens downward because the coefficient of $x^2$ is negative. To find the $x$-value that maximizes profit, we use the formula for the vertex of a parabola:

$x = -\frac{b}{2a}$

Here, $a = -4$ and $b = 20$. Substitute these values: $x = -\frac{20}{2(-4)}$ $x = \frac{20}{8} = 2.5$

Step 3: Verify the result.

The value of $x = 2.5$ maximizes the profit function $P(x)$.

Step 4: Interpret the result.

The maximum profit occurs when $x = 2.5$.

Would you like me to calculate the maximum profit or analyze further?

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5 Related Questions:

1. How do you compute the maximum profit for this function?
2. What happens to profit as $x$ increases beyond the maximizing value?
3. Can you find the breakeven points (values of $x$ where profit equals zero)?
4. How do changes in the revenue or cost functions affect the profit-maximizing value of $x$?
5. What is the significance of a negative coefficient for $x^2$ in the profit function?

Tip:

The vertex formula $x = -\frac{b}{2a}$ is a quick and reliable way to find the maximum or minimum of any quadratic function. Always check the sign of $a$ to determine the direction of the parabola.