In Silicon Valley, a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers. A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination. Suppose that the required monetary pool (in millions of dollars) is given by the following function, where $$ x $$ is the percentage (expressed as a decimal fraction) of the total contaminant removed. Complete parts (A) through (E). $$ P(x)=\frac{11 x}{1-x} \quad 0 \leq x

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In Silicon Valley, a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers. A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination. Suppose that the required monetary pool (in millions of dollars) is given by the following function, where $$ x $$ is the percentage (expressed as a decimal fraction) of the total contaminant removed. Complete parts (A) through (E). $$ P(x)=\frac{11 x}{1-x} \quad 0 \leq x<1 $$ B. $$ P(x) $$ is increasing on $$ [0,1) $$. $$ P(x) $$ is decreasing on C. Where is the graph of $$ P $$ concave upward? Downward? Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) A. $$ P(x) $$ is concave upward on $$ \square $$ and concave downward on B. $$ P(x) $$ is concave upward on $$ \square $$ : C. $$ P(x) $$ is concave downward on $$ \square $$.

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To analyze the function $$ P(x) = \frac{11x}{1-x} $$, we will follow these steps: 1. **Determine where $$ P(x) $$ is increasing or decreasing.** 2. **Find the second derivative to determine concavity.** ### Step 1: Determine where $$ P(x) $$ is increasing or decreasing To find where $$ P(x) $$ is increasing or decreasing, we need to find the first derivative $$ P'(x) $$ and analyze its sign. Using the quotient rule for differentiation, we have: $$ P'(x) = \frac{(1-x)(11) - 11x(-1)}{(1-x)^2} = \frac{11(1-x) + 11x}{(1-x)^2} = \frac{11}{(1-x)^2} $$ Since $$ (1-x)^2 > 0 $$ for $$ 0 \leq x < 1 $$, $$ P'(x) > 0 $$ for all $$ x $$ in this interval. Therefore, $$ P(x) $$ is increasing on $$ [0, 1) $$. ### Step 2: Find the second derivative to determine concavity Next, we find the second derivative $$ P''(x) $$: Using the quotient rule again on $$ P'(x) $$: $$ P''(x) = \frac{(0)(1-x)^2 - 11(2(1-x)(-1))}{(1-x)^4} = \frac{22(1-x)}{(1-x)^4} = \frac{22}{(1-x)^3} $$ Now, we analyze the sign of $$ P''(x) $$: - $$ P''(x) > 0 $$ when $$ 1-x > 0 $$ (which is true for $$ 0 \leq x < 1 $$). Thus, $$ P(x) $$ is concave upward on $$ [0, 1) $$. Since $$ P''(x) $$ is always positive in the interval $$ [0, 1) $$, there are no intervals where $$ P(x) $$ is concave downward. ### Final Answers A. $$ P(x) $$ is concave upward on $$ [0, 1) $$ and concave downward on $$ \emptyset $$. B. $$ P(x) $$ is concave upward on $$ [0, 1) $$. C. $$ P(x) $$ is concave downward on $$ \emptyset $$. $$ P(x) $$ is concave upward on $$ [0, 1) $$ and concave downward on no intervals.

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