1. **Problem statement:** Find all local extrema, the global maximum, and the global minimum of the function \(f(x) = 4 - |x - 3|\) on the domain \([-5, 5]\). 2. **Understand the function:** The function is \(f(x) = 4 - |x - 3|\). This is a V-shaped graph shifted right by 3 and vertically shifted up by 4. 3. **Find critical points:** The absolute value function \(|x - 3|\) is not differentiable at \(x = 3\), so this is a candidate for a local extremum. 4. **Analyze intervals:** - For \(x < 3\), \(|x - 3| = 3 - x\), so \(f(x) = 4 - (3 - x) = 1 + x\). - For \(x > 3\), \(|x - 3| = x - 3\), so \(f(x) = 4 - (x - 3) = 7 - x\). 5. **Check derivatives on each interval:** - For \(x < 3\), \(f'(x) = 1\) (positive slope). - For \(x > 3\), \(f'(x) = -1\) (negative slope). 6. **Determine local extrema:** - At \(x = 3\), the slope changes from positive to negative, indicating a local maximum. 7. **Evaluate function at critical point and endpoints:** - \(f(3) = 4 - |3 - 3| = 4 - 0 = 4\) (local maximum). - \(f(-5) = 4 - |-5 - 3| = 4 - 8 = -4\). - \(f(5) = 4 - |5 - 3| = 4 - 2 = 2\). 8. **Global extrema:** - The global maximum is \(4\) at \(x = 3\). - The global minimum is \(-4\) at \(x = -5\). **Final answer:** - Local maximum at \(x = 3\) with value \(4\). - Global maximum is \(4\) at \(x = 3\). - Global minimum is \(-4\) at \(x = -5\).