Problem

Find the upper and lower bounds of the function \(f(x) = x^2 - 4x + 4\) on the interval [0, 3].

Solution

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.

From Solvely APP
Source: https://solvelyapp.com/problems/uaTjQTvNFW/

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