Applications of Derivatives
Maxima and Minima with Constraints
Grade 12
Question:
<p>The possible value of <i>|a - b|</i>, if <i>a</i><sup>2</sup> + 2<i>b</i><sup>2</sup> is maximum, is given by</p>
<p>(a) 1</p>
<p>(b) 0</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Apply the constraint from function values at specific points to bound |a - b|.
<p>From the constraint |f(x)| ≤ 1 for |x| ≤ 1:</p><p>|f(1) - f(0)| ≤ 2 gives |a + b + c - c| ≤ 2, so |a + b| ≤ 2</p><p>|f(-1) - f(0)| ≤ 2 gives |a - b + c - c| ≤ 2, so |a - b| ≤ 2</p><p>When a² + 2b² is maximized subject to these constraints, we get |a - b| = 2</p><p>∴ Answer is (c) 2.</p>
Correct Answer: c