To determine the price that Bella should sell her book to maximize her revenue, given the revenue function \( y = -14x^2 + 6x \), we need to find the vertex of the parabola represented by this quadratic equation.
The vertex \( x \) of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In our case, \( a = -14 \) and \( b = 6 \). Plugging these values into the formula gives:
\[ x = -\frac{6}{2(-14)} = -\frac{6}{-28} = \frac{6}{28} = \frac{3}{14} \]
Now, we can compute the maximum revenue by substituting this price back into the revenue equation. However, since both the options provided in the question are whole numbers and \( \frac{3}{14} \) doesn't match any of the provided options, we can also recognize that we should check when the function reaches its maximum revenue in practical scenarios (i.e., looking for integer values).
To find the integer price that maximizes revenue, we can calculate the revenue for the provided options:
- For \( x = 6 \): \[ y = -14(6^2) + 6(6) = -14(36) + 36 = -504 + 36 = -468 \]
- For \( x = 12 \): \[ y = -14(12^2) + 6(12) = -14(144) + 72 = -2016 + 72 = -1944 \]
- For \( x = 24 \): \[ y = -14(24^2) + 6(24) = -14(576) + 144 = -8064 + 144 = -7920 \]
- For \( x = 36 \): \[ y = -14(36^2) + 6(36) = -14(1296) + 216 = -18144 + 216 = -17928 \]
From these calculations, it appears that the revenue decreases as we increase the price. The optimal price based on the given options, staying close to the maximum revenue point near \( \frac{3}{14} \), would be $6 since this generates the least loss based on the function trends.
Thus, the price Bella should sell the book to earn maximum revenue, given the options, is:
$6
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