To determine the solution set for the function
[tex]\[ f(x) = 2(x-1)^2+4, \][/tex]
we follow these steps:
1. Understand the Form of the Function:
- This function is a quadratic function in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 4 \)[/tex].
- This is a standard form of a parabola.
2. Identify Key Characteristics:
- The coefficient of the squared term ([tex]\(2\)[/tex]) is positive, indicating that the parabola opens upwards.
- The vertex of this parabola is at the point [tex]\((h, k) = (1, 4)\)[/tex]. This is the lowest point of the parabola since it opens upwards.
3. Determine the Behavior of the Parabola:
- As the vertex is the minimum point and the vertex's y-coordinate is 4, which is greater than zero, the parabola is entirely above the x-axis.
- For a function to have real roots, its graph must intersect the x-axis. Since this parabola opens upwards and its vertex is above the x-axis, it never intersects the x-axis.
4. Conclusion about the Solutions:
- The function [tex]\( f(x) = 2(x - 1)^2 + 4 \)[/tex] does not have any points where [tex]\( f(x) = 0 \)[/tex] for real x-values because the minimum value of [tex]\( f(x) \)[/tex] is 4, which is greater than 0.
- In the context of quadratic equations, if a quadratic function does not intersect the x-axis, it means it has no real solutions but has complex solutions (i.e., solutions involving imaginary numbers).
Given this detailed analysis, the solution set for the function [tex]\( f(x) = 2(x-1)^2+4 \)[/tex] includes only complex solutions.
Thus, the correct phrase describing the solution set is:
[tex]\[ \boxed{\text{D. two complex solutions}} \][/tex]
