Question

the manager of a company uses the function shown to model its daily profit based on the price of a product in doll\\( f(x) = (x - 22)(33 - x) \\)\
a. what is the minimum price, in dollars, to avoid a loss?\
\boxed{} dollars\
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b. what is the maximum price, in dollars, to avoid a loss?\
\boxed{} dollars\
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c. what is the price, in dollars, that results in the greatest profit?\
\boxed{} dollars

Explanation:

Step1: Understand profit and loss condition

To avoid loss, profit \( f(x) \geq 0 \). The function is \( f(x)=(x - 22)(33 - x) \). A product is non - negative when each factor is non - negative or both are non - positive. But since \( 33-x\) and \(x - 22\): when \(x-22\geq0\) and \(33 - x\geq0\), or \(x - 22\leq0\) and \(33 - x\leq0\). The second case \(x-22\leq0\) and \(33 - x\leq0\) implies \(x\leq22\) and \(x\geq33\), which has no solution. So we consider \(x - 22\geq0\) and \(33 - x\geq0\) or find the roots of \(f(x) = 0\) (where profit is zero, and between the roots profit is positive).

Step2: Find roots of \(f(x)=0\)

Set \(f(x)=(x - 22)(33 - x)=0\). Using the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\).

• If \(x-22=0\), then \(x = 22\).
• If \(33 - x=0\), then \(x=33\).

Step3: Analyze part A (minimum price to avoid loss)

The function \(y=(x - 22)(33 - x)=-x^{2}+33x + 22x-726=-x^{2}+55x - 726\) is a quadratic function with \(a=-1<0\), so the parabola opens downwards. The profit is non - negative (no loss) between the two roots \(x = 22\) and \(x = 33\). So the minimum price to avoid loss is when the left - most root, which is \(x = 22\).

Step4: Analyze part B (maximum price to avoid loss)

From the roots of \(f(x)=(x - 22)(33 - x)=0\) ( \(x = 22\) and \(x = 33\) ), and since the parabola opens downwards, the profit is non - negative between \(x = 22\) and \(x = 33\). So the maximum price to avoid loss is the right - most root, \(x=33\).

Step5: Analyze part C (price for greatest profit)

For a quadratic function \(y = ax^{2}+bx + c\) (\(a
eq0\)), the x - coordinate of the vertex (which gives the maximum for \(a<0\)) is \(x=-\frac{b}{2a}\). First, expand \(f(x)=(x - 22)(33 - x)=-x^{2}+55x - 726\). Here \(a=-1\), \(b = 55\).
Using the formula \(x=-\frac{b}{2a}=-\frac{55}{2\times(-1)}=\frac{55}{2}=27.5\). We can also note that for a quadratic function in factored form \(y=(x - r_1)(r_2 - x)\) (where \(r_1\) and \(r_2\) are roots), the x - coordinate of the vertex is the mid - point of the roots. The mid - point of \(x_1 = 22\) and \(x_2=33\) is \(\frac{22 + 33}{2}=\frac{55}{2}=27.5\).

Answer:

A. 22 dollars
B. 33 dollars
C. 27.5 dollars