Applications of Derivatives
Monotonicity and Extrema
Grade 12
Question:
<p>If <i>a</i> < <i>b</i> and <i>f</i>(<i>x</i>) = |<i>x</i> − <i>a</i>| + |<i>x</i> − <i>b</i>|, <i>x</i> ∈ ℝ, then the minimum value of <i>f</i>(<i>x</i>) is</p>
<p>(A) 0</p>
<p>(B) |<i>a</i> − <i>b</i>|</p>
<p>(C) <i>a</i> + <i>b</i></p>
<p>(D) Cannot be determined</p>
Step-by-Step Solution
Key Concept: The sum of distances from a point to two fixed points is minimized when the point lies between them. On the interval [a, b], both absolute value terms combine to give the constant distance |a − b|.
<p><strong>Step 1:</strong> The function <i>f</i>(<i>x</i>) = |<i>x</i> − <i>a</i>| + |<i>x</i> − <i>b</i>| with <i>a</i> < <i>b</i>.</p><p><strong>Step 2:</strong> Analyzing the function: <i>f</i>(<i>x</i>) is decreasing in (−∞, <i>a</i>], constant in [<i>a</i>, <i>b</i>], and increasing in [<i>b</i>, ∞).</p><p><strong>Step 3:</strong> The constant value on [<i>a</i>, <i>b</i>] is |<i>a</i> − <i>b</i>|.</p><p>∴ The minimum value of <i>f</i>(<i>x</i>) is |<i>a</i> − <i>b</i>|.</p>
Correct Answer: B