Given revenue    and    cost  the functions
R(x)=-4x^2+31x             C(x)=11x+29  
determine the value of x that maximizes profit

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.

Given revenue and cost functions R(x) = -4x^2 + 31x and C(x) = 11x + 29, determine the value of x that maximizes profit.

Learn how to maximize profit by analyzing revenue and cost functions: R(x) = -4x^2 + 31x and C(x) = 11x + 29. This solution uses algebraic methods, including the vertex formula, to find the profit-maximizing value of x. Perfect for Grades 10-12 students studying quadratic equations.

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