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). \[ P(x)=\frac{11 x}{1-x} \quad 0 \leq x<1 \] (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 \( [0,1) \). C. \( P(x) \) is concave downward on (C) Find any horizontal and vertical asymptote. Find the vertical asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The vertical asymptote(s) is (are) \( x= \) \( \square \) (Simplify your answer. Use a comma to separate answers as needed) B. There are no vertical asymptotes.
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Beyond the Answer
To dive into \( P(x) \), let's start with its concavity. The second derivative, \( P''(x) \), helps us analyze the concavity. For \( 0 \leq x < 1 \), the function is concave upward because \( P''(x) > 0 \). So, on this interval, it’s always bending upwards like a smile! Therefore, \( P(x) \) is concave downward only at the endpoint, so \( P(x) \) is just slightly sad at 1, as it can never quite reach there. Now, let’s tackle asymptotes! The function has a vertical asymptote where the denominator approaches zero. In this case, setting \( 1 - x = 0 \), we find that \( x = 1 \) is the vertical asymptote. For horizontal asymptotes, we look at the behavior as \( x \) approaches infinity, but since \( x \) only goes up to 1 in our scenario, we conclude there are no horizontal asymptotes here—it's just all about that vertical one!
