For the function <span class="math-inline">f(x) = |x + 3| - |x + 1| - |x - 1| + |x - 3|</span>, identify correct option(s)
Range of f(x) is (-\infty, 4]
maximum value of f(x) is 4
f(x) = 4 has infinite solutions
f(x) = 0 has infinite solutions
Step-by-Step Solution
Key Concept: Analyze the piecewise function by identifying critical points where absolute value expressions change sign, then evaluate f(x) on each interval to determine its range and behavior.
<p><strong>Step 1: Identify critical points.</strong> The expressions |x+3|, |x+1|, |x-1|, |x-3| change sign at x = -3, -1, 1, 3. These divide the real line into 5 intervals.</p><p><strong>Step 2: Analyze f(x) on each interval.</strong></p><p><u>For x ≤ -3:</u> f(x) = -(x+3) - (-(x+1)) - (-(x-1)) + (-(x-3)) = -x-3 + x+1 + x-1 - x+3 = 0</p><p><u>For -3 < x ≤ -1:</u> f(x) = (x+3) - (-(x+1)) - (-(x-1)) + (-(x-3)) = x+3 + x+1 + x-1 - x+3 = 2x+6. Range: (0, 4]</p><p><u>For -1 < x ≤ 1:</u> f(x) = (x+3) - (x+1) - (-(x-1)) + (-(x-3)) = x+3 - x-1 + x-1 - x+3 = 4</p><p><u>For 1 < x ≤ 3:</u> f(x) = (x+3) - (x+1) - (x-1) + (-(x-3)) = x+3 - x-1 - x+1 - x+3 = -2x+6. Range: [0, 4)</p><p><u>For x > 3:</u> f(x) = (x+3) - (x+1) - (x-1) + (x-3) = x+3 - x-1 - x+1 + x-3 = 0</p><p><strong>Step 3: Determine range.</strong> From the intervals: f(x) = 0 for x ≤ -3 and x > 3; f(x) ∈ (0,4] for -3 < x ≤ -1; f(x) = 4 for -1 < x ≤ 1; f(x) ∈ [0,4) for 1 < x ≤ 3. Overall range: [0, 4]</p><p><strong>Step 4: Check each option.</strong></p><p><strong>Option A:</strong> Range is [0, 4], not (-∞, 4]. FALSE</p><p><strong>Option B:</strong> Maximum value is 4 (attained on [-1, 1]). TRUE</p><p><strong>Option C:</strong> f(x) = 4 for all x ∈ [-1, 1], which is an infinite set of solutions. TRUE</p><p><strong>Option D:</strong> f(x) = 0 for x ∈ (-∞, -3] ∪ (3, ∞), which is an infinite set of solutions. TRUE</p><p><strong>∴ Answer:</strong> B,C,D</p>
Correct Answer: B,C,D