Prove that the function \(f(x) = x^2 - 4x + 4\) has at least one root in the interval \([1, 2]\).
Answer
Step 4: Since \(f(1) > 0\) and \(f(2) = 0\), by the Intermediate Value Theorem, there exists at least one \(c\) in \([1, 2]\) such that \(f(c) = 0\). Therefore, the function \(f(x) = x^2 - 4x + 4\) has at least one root in the interval \([1, 2]\).
Step 1 ：Step 1: We first calculate the function values at the endpoints of the interval, i.e., \(f(1)\) and \(f(2)\).
Step 2 ：Step 2: \(f(1) = (1)^2 - 4(1) + 4 = 1 - 4 + 4 = 1\)
Step 3 ：Step 3: \(f(2) = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0\)
Step 4 ：Step 4: Since \(f(1) > 0\) and \(f(2) = 0\), by the Intermediate Value Theorem, there exists at least one \(c\) in \([1, 2]\) such that \(f(c) = 0\). Therefore, the function \(f(x) = x^2 - 4x + 4\) has at least one root in the interval \([1, 2]\).
