Final answer:
The function f(x) = x² - 4x + 5 has a global minimum at x = 2 with a minimum value of 1, because it is a parabola opening upwards with its vertex at the boundary of the region x > 2.
Step-by-step explanation:
To explain why the function f(x) = x² - 4x + 5 must have a global minimum at some point in the region where x > 2, we first need to recognize that the function is a parabola that opens upwards, since the coefficient of x² is positive. Because it is a continuous and differentiable function, we can find critical points where the derivative of the function equals zero.
First, take the derivative of the function: f'(x) = 2x - 4. Setting this equal to zero and solving for x, we get:
f'(x) = 0
2x - 4 = 0
x = 2
Since this is the only critical point and the function is unbounded and opens upwards, x = 2 must be the point of the global minimum within the region x > 2. At x = 2, the value of f(x) is:
f(2) = 2² - 4(2) + 5 = 4 - 8 + 5 = 1.
Therefore, the global minimum value of f(x) in the region x > 2 is 1. This analysis relies on the understanding that a parabola opening upwards will have its vertex as the lowest point on the graph. Since the critical point x = 2 lies at the boundary of our region of interest and f(x) is going to increase for all x > 2, it is indeed the global minimum in the region where x > 2.