To find the value of k that makes the equation f(x) = 1/2 have at least two solutions in the interval [0,2], we need to analyze the behavior of the function on that interval based on the given table.
First, let's consider the given values of f(x) in the table. At x = 0, f(x) = 1, and at x = 2, f(x) = 2. We also know that f(x) is continuous on the closed interval [0, 2].
Since f(x) is continuous on [0, 2], and the function values at the endpoints of the interval are different (f(0) = 1 and f(2) = 2), by the Intermediate Value Theorem, there must exist at least one value c in the open interval (0, 2) such that f(c) = 1/2. This means that the equation f(x) = 1/2 has at least one solution in the interval (0, 2).
But we want to find the value of k that makes the equation have at least two solutions in the interval [0, 2]. For that to happen, the function f(x) must have at least one more value between 0 and 2 where f(x) = 1/2. Since the function values at the endpoints are different, f(x) must pass through 1/2 somewhere between 0 and 2.
From the given table, we see that f(x) has the value k at x = 1. So, for f(x) to have at least two solutions, k must be equal to 1/2. This is because when k = 1/2, the function f(x) will have a value of 1/2 at x = 1 as well (which is between 0 and 2), giving us the second solution of the equation f(x) = 1/2 in the interval [0, 2].
Therefore, the value of k that makes the equation f(x) = 1/2 have at least two solutions in the interval [0, 2] is k = 1/2.