Find the upper and lower bounds of the function \(f(x) = x^2 - 4x + 4\) on the interval [0, 3].
Answer
Step 3: The upper bound of the function on the interval is the maximum of these values, and the lower bound is the minimum. So, the upper bound is 4 and the lower bound is 0.
Step 1 ：Step 1: Identify the critical points in the interval. The critical points are found by taking the derivative of the function and setting it equal to zero. So, \(f'(x) = 2x - 4 = 0\). Solving for \(x\), we get \(x = 2\).
Step 2 ：Step 2: Evaluate the function at the critical point and at the endpoints of the interval. So we find \(f(0) = 0^2 - 4*0 + 4 = 4\), \(f(2) = 2^2 - 4*2 + 4 = 0\), and \(f(3) = 3^2 - 4*3 + 4 = 1\).
Step 3 ：Step 3: The upper bound of the function on the interval is the maximum of these values, and the lower bound is the minimum. So, the upper bound is 4 and the lower bound is 0.
