Question

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Explanation:

Identify the quadratic function

Using the Quadratic Function Definition knowledge point

$$ P(x) = -4x^2 + 32x - 39 $$

The coefficients are \(a = -4\), \(b = 32\), and \(c = -39\).

Find the vertex x-coordinate

To find the number of pairs of shoes \(x\) that maximizes the profit, we calculate the x-coordinate of the vertex of the parabola.

$$ x = -\frac{b}{2a} = -\frac{32}{2(-4)} = \frac{-32}{-8} = 4 $$

Verify the maximum profit

Since the leading coefficient \(a = -4\) is negative (\(a < 0\)), the parabola opens downward, confirming that the vertex represents a maximum.

$$ P(4) = -4(4)^2 + 32(4) - 39 = -64 + 128 - 39 = 25 $$

The maximum profit is 25 units when selling 4 pairs of shoes.

Answer:

• 6 pairs of shoes
• 7 pairs of shoes
• 5 pairs of shoes
• 4 pairs of shoes (Correct answer)