The stock price for a certain corporation can be approximated for the years 2010 through 2017 by \( f(x)=6.6 x^{2}-200 x+1612 \), where \( x=10 \) corresponds to the year 2010. During what year did a local minimum stock price occur?

To find the year when a local minimum stock price occurred, we need to analyze the function \( f(x) = 6.6x^2 - 200x + 1612 \). Since this is a quadratic function, it will have a parabolic shape. The local minimum occurs at the vertex of the parabola. The x-coordinate of the vertex for a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] In our case, \( a = 6.6 \) and \( b = -200 \). Let's calculate the x-coordinate of the vertex. 1. Calculate \( -b \): \[ -b = -(-200) = 200 \] 2. Calculate \( 2a \): \[ 2a = 2 \times 6.6 = 13.2 \] 3. Now, substitute these values into the vertex formula: \[ x = \frac{200}{13.2} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{200}{13.2}\) - step1: Convert the expressions: \(\frac{200}{\frac{66}{5}}\) - step2: Multiply by the reciprocal: \(200\times \frac{5}{66}\) - step3: Reduce the numbers: \(100\times \frac{5}{33}\) - step4: Multiply: \(\frac{100\times 5}{33}\) - step5: Multiply: \(\frac{500}{33}\) The calculation gives us: \[ x = \frac{200}{13.2} \approx 15.15 \] This means that the local minimum stock price occurs at approximately \( x = 15.15 \). Since \( x = 10 \) corresponds to the year 2010, we can find the corresponding year by adding this value to 2000: \[ \text{Year} = 2000 + 15.15 \approx 2015.15 \] Since we are looking for the year, we round down to the nearest whole number, which gives us 2015. Therefore, the local minimum stock price occurred in the year **2015**.