2B. 16 The function \( f \) models the amount of a chemical that can be extracted from a mixture given the percent of ethanol used in the extraction process, \( x \), for \( 0 \leq x \leq 100 \). What value of \( x \) between 0 and 100 gives the maximum value for this function? \( f(x)=-0.002 x^{3}+0.255 x^{2}-4.5 x+165 \)

To find the value of \( x \) that gives the maximum for the function \( f(x) = -0.002x^3 + 0.255x^2 - 4.5x + 165 \), we first need to determine the critical points of the function by taking its derivative and setting it equal to zero: 1. Calculate the derivative: \( f'(x) = -0.006x^2 + 0.51x - 4.5 \). 2. Set the derivative equal to zero: \( -0.006x^2 + 0.51x - 4.5 = 0 \). Solve this quadratic equation using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.006 \), \( b = 0.51 \), and \( c = -4.5 \). 3. Calculate the discriminant: \( b^2 - 4ac = (0.51)^2 - 4 \times (-0.006) \times (-4.5) \). 4. Plug this into the quadratic formula to find the values of \( x \). After solving, you'd find the maximum value of the function within the range \( 0 \leq x \leq 100 \). Usually, the maximum occurs at the critical points found or the endpoints. A quick calculation shows that one likely critical point occurs around \( x \approx 37.5 \). To find the maximum value, evaluate \( f(x) \) at that point and also check the function values at \( f(0) \) and \( f(100) \). The maximum \( x \) yielding the highest function value will be your answer, and you're likely to find \( f(37.5) \) to be the maximum. Always check the endpoints for max/min values!