Final answer:
The quadratic function f(x) = 2x² - 3x + 1 has a minimum value, not listed in the given options, at (3/4, -1/8) because the coefficient of x² is positive, indicating the parabola opens upwards. There is no maximum value.
Step-by-step explanation:
To find the maximum or minimum value of the quadratic function f(x) = 2x² - 3x + 1, we need to use the vertex form of the function, which gives us the coordinates of the vertex of the parabola represented by the function. The vertex form is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Since the coefficient a in front of x² is positive (a = 2), the parabola opens upwards, and therefore has a minimum value but no maximum.
To find the vertex, we use the vertex formula h = -b/(2a) and k = f(h). For our function:
- a = 2
- b = -3
- h = -(-3)/(2 * 2) = 3/4
- k = f(3/4) = 2(3/4)² - 3(3/4) + 1
Calculating the value of k, we get:
k = 2(9/16) - 9/4 + 1 = 9/8 - 9/4 + 1 = 9/8 - 18/8 + 8/8 = -1/8
Thus, the vertex of the parabola, which gives us the minimum value, is (3/4, -1/8). There is no maximum value since the parabola opens upwards. The correct answer is (b) Maximum: none, Minimum: -1/8, which is not listed in the options. Possibly there is an error in the question.