Desmond buys and selis shares in the stock market The value of the shares be bought is given by the function \( y=3 x^{2}-5 x+7 \), where \( y \) is the value of the shares in thousands of dollars and \( x \) is the tume io years after it was first bought. (a) What is the mimmum value of the shares and when does it occur?

To find the minimum value of the function \( y = 3x^2 - 5x + 7 \), we can use the vertex formula for a quadratic function of the form \( y = ax^2 + bx + c \). The x-coordinate of the vertex, which gives the minimum value when \( a > 0 \), is given by: \[ x = -\frac{b}{2a} \] In this case, \( a = 3 \) and \( b = -5 \). Plugging in these values: \[ x = -\frac{-5}{2 \cdot 3} = \frac{5}{6} \] Now, we can find the minimum value of \( y \) by substituting \( x = \frac{5}{6} \) back into the function: \[ y = 3\left(\frac{5}{6}\right)^2 - 5\left(\frac{5}{6}\right) + 7 \] Calculating \( \left(\frac{5}{6}\right)^2 \): \[ \left(\frac{5}{6}\right)^2 = \frac{25}{36} \] Now substituting this into the equation: \[ y = 3 \cdot \frac{25}{36} - 5 \cdot \frac{5}{6} + 7 \] Calculating each term: 1. \( 3 \cdot \frac{25}{36} = \frac{75}{36} \) 2. \( 5 \cdot \frac{5}{6} = \frac{25}{6} = \frac{150}{36} \) (converting to a common denominator) 3. \( 7 = \frac{252}{36} \) (also converting to a common denominator) Now, combining these: \[ y = \frac{75}{36} - \frac{150}{36} + \frac{252}{36} \] Combining the fractions: \[ y = \frac{75 - 150 + 252}{36} = \frac{75 + 252 - 150}{36} = \frac{177}{36} \] Now simplifying \( \frac{177}{36} \): \[ \frac{177}{36} = 4.9167 \text{ (approximately)} \] Thus, the minimum value of the shares occurs at \( x = \frac{5}{6} \) years, and the minimum value of the shares is approximately \( 4.92 \) thousand dollars. **Final Answer:** The minimum value of the shares is approximately \( 4.92 \) thousand dollars, occurring at \( x = \frac{5}{6} \) years.