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An absolute minimum value of a function is the lowest value the function can take on a given interval. In more mathematical terms, for a function f(x), if there exists a point x=c in the interval [a, b] such that f(c) ≤ f(x) for all other x values in the interval, then f(c) is the absolute minimum value of the function on that interval.

Under the Extreme Value Theorem, a continuous function on a closed interval [a, b] must have an absolute minimum value and an absolute maximum value. These extremal values occur at critical points (where the derivative is zero or does not exist) or the interval's endpoints. Therefore, a function can have an absolute minimum value at one of the endpoints if the value at that endpoint is less than or equal to the function's values at any other critical points and at the other endpoint.

Let's consider a quadratic function f(x) = (x - 1)^2 on the closed interval [0, 2]. This function is continuous on this interval, so it has an absolute minimum value according to the Extreme Value Theorem. First, we find the critical points by taking the derivative of the function and finding where it is zero or does not exist. f'(x) = 2(x - 1) Setting f'(x) to 0, we find: 0 = 2(x - 1) x = 1 There is one critical point at x=1. Now, we compare the function values at the critical point and the endpoints: f(0) = (0 - 1)^2 = 1 f(1) = (1 - 1)^2 = 0 f(2) = (2 - 1)^2 = 1 Since the function value at the critical point x=1 (f(1)=0) is the lowest compared to the endpoints values f(0) = 1 and f(2) = 1, the function has an absolute minimum value at the point (1, 0).